Example Mechanics Problems

On this page I give lots of examples of solved mechanics problems. They are provided as handwritten PDF files which you can conveniently download. The examples cover mechanics topics at a level typical of physics courses in senior high school, first year college/university, and upper year college/university. This will be very useful when studying for tests and exams. The aim here is to cover many different types of problems, from easy to difficult, to help you study and prepare more effectively. I know that getting access to lots of good problems and their solutions can sometimes be difficult, which is why I created this page, so you have a lot of problems and their solutions all in one convenient place. All these example problems are my own creation, so you won't find an exact copy of them anywhere else. In order to keep them relevant to your courses, I have referenced commonly used physics books to get ideas and inspiration for problems. So there will of course be similarities between them and the problems you will encounter in your classes and physics books.

The example problems are all uniquely numbered for easy reference. The problem statement is given, and then the solution is provided as a PDF file which you can download.

If you want to review the theory behind all these problems, I conveniently cover that on this website, in the kinematics and dynamics pages. Note that this page is a work in progress and new content will be added regularly.

To see the example problems click on the category you are interested in:

Kinematics – 1-D problems involving free-fall acceleration (motion along a straight line) – Senior high school and first year college/university
Kinematics – 1-D problems involving constant acceleration (motion along a straight line) – Senior high school and first year college/university
Kinematics – 1-D problems involving average velocity and average speed (motion along a straight line) – Senior high school and first year college/university
Kinematics – 1-D problems involving instantaneous velocity and speed (motion along a straight line) – Senior high school and first year college/university
Kinematics – 1-D problems involving average acceleration and instantaneous acceleration (motion along a straight line) – Senior high school and first year college/university
Kinematics – 2-D and 3-D problems involving position and displacement – Senior high school
Kinematics – 2-D and 3-D problems involving instantaneous velocity, average velocity, and average speed – Senior high school and first year college/university
Kinematics – 2-D and 3-D problems involving instantaneous acceleration and average acceleration – Senior high school and first year college/university
Kinematics – Projectile motion problems – Senior high school and first year college/university
Kinematics – Uniform circular motion – Senior high school and first year college/university
Kinematics – 1-D problems involving relative motion – Senior high school and first year college/university
Kinematics – 2-D problems involving relative motion – Senior high school and first year college/university


Kinematics – 1-D problems involving free-fall acceleration (motion along a straight line) – Senior high school and first year college/university

Problem # 1:

A building is under construction, and a construction worker is standing on top of a 130 m high elevator shaft. The worker accidentally drops his hammer down the shaft.

(a) At what speed does the hammer hit the ground?

(b) How much time passes between when the hammer is dropped and when it hits the ground?

(c) What fraction of the total airborne time does the hammer spend in the top 75% of the falling distance?

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Problem # 2:

A model rocket is launched vertically, and has a constant acceleration of 5.0 m/s2 for 8.0 seconds, after which there is no fuel left.

(a) What is the maximum height reached by the rocket?

(b) How much time passes between when the rocket is launched and when it lands?

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Problem # 3:

An object is dropped from rest, and one second before it lands it is at half its initial drop height.

(a) What is the falling time?

(b) What is the drop height?

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Problem # 4:

A ball is thrown vertically upward. On its way up it passes point A at a speed v. On its way down it passes point B at a speed that is (3/4)v. Point B is 4.5 m higher than point A.

(a) What is the speed v?

(b) What is the distance between point A and the peak height reached by the ball?

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Problem # 5:

A drop tower at an amusement park rises at 5 m/s and is 45 m above the ground when one of the riders drops her phone.

(a) How long does it take for the phone to fall to the ground?

(b) At what speed does the phone hit the ground?

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Problem # 6:

A girl is standing in an elevator moving upward at 5 m/s. She places a launch toy on the floor of the elevator, which then launches a ball straight up at 5.5 m/s relative to the elevator. The girl catches the ball 1.0 seconds later. At the instant the ball is caught, the floor of the elevator is 32 m above the ground.

(a) What is the height of the ball above the ground at the instant it is caught?

(b) What is the height of the elevator floor above the ground at the instant the ball is launched?

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Problem # 7:

A child is standing in an elevator with glass walls, at a mall. She throws a ball in the air at a vertical upward speed of 4.5 m/s relative to the elevator, and from a height of 1.3 m relative to the elevator floor. At the same time, the elevator is moving upward at 3 m/s, starting from ground level.

(a) From the perspective of the child, what is the maximum height reached by the ball?

(b) From the perspective of someone in the mall (outside the elevator), what is the maximum height reached by the ball?

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Problem # 8:

A mischievous student drops an egg from the window of his dorm room. The egg falls straight down onto the hood of a car parked below. A few floors below, someone is recording a video on their webcam, which is facing the window. The egg is recorded falling past the window. The person recording the video is a physics student, and she sees an opportunity to solve an interesting physics problem while also determining the height, and consequently the room, that the egg was dropped from. She analyzes the video, and determines that it took the egg 0.14 seconds to fall from the top of the window to the bottom. She then measures the height of the window to be 1.30 meters. From what height, measured from the top of the window, was the egg dropped?

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Problem # 9:

A game is played by two children, in which one child, at a height of 10 m above the ground, drops a rock with no initial speed. The second child also drops a rock, from a height of 15 m above the ground. The second child drops the rock tr seconds after the first child drops the rock, with an initial downward speed of 10 m/s. What is the value of tr so that both rocks hit the ground at the same time?

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Here are additional free fall problems for you to work on.

Kinematics – 1-D problems involving constant acceleration (motion along a straight line) – Senior high school and first year college/university

Problem # A-1:

A car travelling on a straight road at 25 m/s undergoes constant acceleration until it reaches a speed of 40 m/s. The car then maintains this speed for 6.0 seconds. The brakes are then applied, causing the car to undergo constant deceleration until it once more reaches a speed of 25 m/s. If it takes the car 25 seconds from the time that it starts accelerating to the time that it slows down to 25 m/s, how far does it travel in this time?

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Problem # A-2:

mechanics example prob kin a2

A train is moving at high speed on a straight track, while at the same time a locomotive is moving in the opposite direction on the same track. In order to avoid a collision, the locomotive must move on to the siding before a collision becomes unavoidable. At the same time, the train must decelerate by putting the brakes on. At the instant shown, the front of the train is 0.35 mi from the back of the locomotive, the back of the locomotive is 0.1 mi from the siding entrance, the speed of the train is 80 mi/h, and the maximum speed of the locomotive is 20 mi/h. What is the minimum deceleration of the train?

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Problem # A-3:

mechanics example prob kin a3

Given the above graph of a(t), sketch the graph of v(t) and d(t).

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Problem # A-4:

The takeoff speed of a commercial jet is 260 km/h. If the runway is 2.1 km long, what is the minimum constant acceleration of the jet?

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Problem # A-5:

A truck driving on a paved road is capable of decelerating at a constant value of 5 m/s2.

(a) If the truck is initially travelling at 27.4 m/s, how long does it take to come to a complete stop?

(b) How far does the truck travel in this time?

(c) Sketch a graph of distance vs. time and speed vs. time, when the brakes are applied.

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Problem # A-6:

A distracted driver is cruising at 25 m/s when she suddenly notices a truck directly ahead moving at 15 m/s, in the same direction. At the instant she brakes, the distance between the front of the car and the back of the truck is d. If the car decelerates at 5.0 m/s2, and the truck maintains its speed, what is the minimum value of d in order to avoid a collision?

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Problem # A-7:

A distracted driver is cruising at 25 m/s when he suddenly notices a truck directly ahead moving at 15 m/s, in the same direction. At the instant he brakes, the distance between the front of the car and the back of the truck is 15 m. If the car decelerates at ac, and the truck maintains its speed, what is the minimum value of ac in order to avoid a collision?

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Problem # A-8:

A distracted driver is cruising at 25 m/s when he suddenly notices a truck directly ahead moving at 15 m/s, in the same direction. At the instant he brakes, the distance between the front of the car and the back of the truck is 9 m. The car decelerates at 5 m/s2, and one second after the driver of the car applies the brakes, the truck driver notices the car behind and starts accelerating at at. What is the minimum value of at in order to avoid a collision?

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Problem # A-9:

A distracted driver is driving on the wrong side of the road, when he notices an oncoming vehicle moving towards him. He quickly applies the brakes, causing his car to decelerate at 4.5 m/s2, from an initial speed of 20 m/s. The other driver also applies the brakes, 1.1 seconds later, causing his car to decelerate at 5 m/s2, from an initial speed of 23 m/s. At the instant the distracted driver applies the brakes, the front of each car is separated by a distance of 117 m. Is there a collision between the two cars?

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Kinematics – 1-D problems involving average velocity and average speed (motion along a straight line) – Senior high school and first year college/university

Problem # B-1:

The speed of sound in air is 330 m/s at 0 degrees Celsius. If the average velocity of a jet plane is 2.3 times the speed of sound, how far does it travel in 0.25 seconds?

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Problem # B-2:

A delivery truck travels up a hill at a constant speed of 50 km/h, in order to deliver a package. After the package is delivered, the truck travels down the same hill at 80 km/h. What is the average speed of the truck for the round trip?

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Problem # B-3:

mechanics example prob kin b3

The graph shown above shows velocity vs. time for a particle moving along a straight line. What is the average velocity and average speed for the particle for the entire time the particle is in motion?

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Problem # B-4:

A particle is moving along the x-axis at a speed defined by s(t) = at + b, where a and b are constants and t is time. If the particle starts moving at time t1 and stops moving at time t2, what is the value of a and b so that the average velocity and average speed, between t1 and t2, is equal to (1/2)(s(t1) + s(t2)).

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Problem # B-5:

The position of an object moving along the x-axis is given by x = -2t3 + 2t2 + 3t − 5, where x is in meters, and t is in seconds.

(a) Find the position of the object at t = 2.0 s, and t = 3.5 s.

(b) What is the displacement of the object between t = 2.0 s, and t = 3.5 s ?

(c) What is the average velocity of the object between t = 2.0 s, and t = 3.5 s ?

(d) What is the average speed of the object between t = 2.0 s, and t = 3.5 s ?

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Problem # B-6:

mechanics example prob kin b6

The position of an object moving along the x-axis is shown above, where x is in meters and t is in seconds.

(a) What is the average velocity of the object between 0 and t3 ?

(b) What is the average speed of the object between 0 and t3 ?

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Problem # B-7:

A delivery truck drives 6.5 km along a straight road. The driver then exits the truck, walks 1.5 km to deliver a package at one house, and then continues walking another 2 km to deliver a package to another house. The driver then walks back to the truck. The driving speed of the truck is 70 km/h, and the walking speed of the driver is 5 km/h.

(a) What is the average velocity and average speed of the driver from the start of the drive until the time that the package is delivered to the second house?

(b) What is the average velocity and average speed of the driver from the start of the drive until the time that the driver returns to the truck?

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Problem # B-8:

An object moves along the positive x-axis at 2 m/s, from point A to point B. It then moves at 3 m/s, from point B to point C. If point C is located halfway between points A and B, what is the average velocity and average speed of the object between points A and C ?

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Problem # B-9:

An object moves along the positive x-axis, to the right, at speed s1, from point A to point B. It then moves at speed s2, from point B to point C. If point C is located to the left of point B, derive an equation relating the average velocity of the object to the average speed of the object, between points A and C.

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Kinematics – 1-D problems involving instantaneous velocity and speed (motion along a straight line) – Senior high school and first year college/university

Problem # C-1:

The position of a particle is given by x = 4 − 10t + 2t2 − 3t3, where t is in seconds and x is in meters.

(a) What is the velocity of the particle at t = 2 s ?

(b) Is the position of the particle increasing or decreasing at t = 2 s ?

(c) What is the speed of the particle at t = 2 s ?

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Problem # C-2:

A particle's position is given by x = 6 + 4t − 12t2 + 6t3, where t is in seconds and x is in meters.

(a) At what time is the velocity of the particle equal to -1.5 m/s ?

(b) At what time is the speed of the particle equal to 1.5 m/s ?

(c) What is the minimum velocity and minimum speed of the particle?

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Kinematics – 1-D problems involving average acceleration and instantaneous acceleration (motion along a straight line) – Senior high school and first year college/university

Problem # D-1:

A particle's position is given by x = 2 + 4t − 5t2 + 9t3, where t is in seconds and x is in meters.

(a) What is the average acceleraton of the particle between t = 1 s and t = 2 s ?

(b) What is the acceleraton of the particle at t = 3 s ?

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Problem # D-2:

A particle is moving towards the right at 21 m/s, at a time of 3.1 s, and is moving towards the left at 18 m/s, at a time of 6.4 s. What is the average acceleration of the particle from 3.1 s to 6.4 s ?

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Problem # D-3:

mechanics example prob kin d3

A particle moves in a straight line, as represented by the above graph of velocity vs. time. Sketch a graph representing the acceleration of this particle.

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Problem # D-4:

mechanics example prob kin d4

A particle moves in a straight line, as represented by the above graph. Sketch a graph representing the acceleration of this particle.

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Problem # D-5:

A particle moves in a straight line at 12 m/s, and some time later it is moving at −21 m/s. If the average acceleration of the particle is −2.5 m/s2, how much time has passed between the initial and final velocity?

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Kinematics – 2-D and 3-D problems involving position and displacement – Senior high school

Problem # E-1:

A particle has the following coordinates: x = −4.0 m, y = 7.0 m, and z = 2.0 m.

(a) Find the position vector in unit-vector notation.

(b) What is the magnitude of this vector?

(c) Sketch the vector on a right-handed coordinate system.

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Problem # E-2:

A particle has the following coordinates initially: x = 4.5 m, y = −3.5 m. Some time later the coordinates are: x = −1.5 m, y = 2.0 m.

(a) Find the displacement vector of the particle.

(b) Sketch the initial, final, and displacement vector of the particle.

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Problem # E-3:

A particle has the following coordinates initially: x = 1.0 m, y = −3.0 m, z = 8.0 m. Some time later the coordinates are: x = 3.0 m, y = −5.0 m, z = 4.0 m. What is the displacement vector of the particle?

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Kinematics – 2-D and 3-D problems involving instantaneous velocity, average velocity, and average speed – Senior high school and first year college/university

Problem # F-1:

A plane flies 350 mi east from airport A to airport B in 50 min, and then flies 500 mi south from airport B to airport C in 1.8 h.

(a) Determine the displacement vector, for the total trip.

(b) Determine the average velocity vector, for the total trip.

(c) Determine the average speed, for the total trip.

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Problem # F-2:

In 4.2 h, a weather balloon drifts 10.2 km west, 18.7 km south, and 3.1 km upward.

(a) Determine the magnitude of the balloon's average velocity, and the angle that this vector makes with the horizontal.

(b) Determine the average speed of the balloon.

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Problem # F-3:

The position of a particle is given by r = 4i − 2t2j + 8t3k, with r in meters, and t in seconds.

(a) What is the instantaneous velocity of the particle at t = 2.0 s ?

(b) What is the magnitude of the instantaneous velocity at t = 2.0 s ?

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Kinematics – 2-D and 3-D problems involving instantaneous acceleration and average acceleration – Senior high school and first year college/university

Problem # G-1:

A particle has initial velocity v = 3.0i − 4.0j + 8.0k, and 5.0 s later it has final velocity v = 2.0i − 1.0j + 3.0k. Units are in m/s.

(a) What is the average acceleration over the 5.0 s interval?

(b) What is the magnitude of the average acceleration, and show the orientation of the average acceleration.

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Problem # G-2:

A particle is initially located at the origin and has an initial velocity of v = 3.0i + 2.0j − 1.0k, in meters per second. It experiences a constant acceleration of a = −1.0i − 1.5j, in meters per second squared.

(a) What is the velocity of the particle when its y-coordinate is a maximum?

(b) Where is the particle located at this instant?

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Problem # G-3:

The position of a particle is given by r = t2i + (t3 − 5t)j − 3.0tk, in meters. What is the instantaneous acceleration of the particle? What is the orientation of the instantaneous acceleration at t = 1.0 s ?

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Kinematics – Projectile motion problems – Senior high school and first year college/university

Problem # H-1:

mechanics example prob kin h1

A ball rolls down a ramp that is inclined at 15° with the horizontal. At the edge of the ramp (point A), the velocity of the ball is v0. Determine the range of values of v0 so that the ball enters the horizontal pipe shown.

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Problem # H-2:

mechanics example prob kin h2

A remote controlled toy car is driven off the edge of a table, at point A, at a speed of 2.2 m/s. It lands at point B. If the table is 1.1 m high, what is the horizontal distance, L, between point A and point B?

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Problem # H-3:

mechanics example prob kin h3

A remote controlled toy car is driven off the edge of a ramp, at point A, at a speed of 3 m/s. It lands at point B. If the edge of the ramp is at a height of 0.8 m, and it is inclined at 20° with the horizontal, what is the horizontal distance, L, between point A and point B?

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Problem # H-4:

mechanics example prob kin h4

A person standing at the top of a hill throws a rock with an initial velocity of 12 m/s at an angle of 20° below the horizontal.

(a) Calculate the horizontal displacement of the rock 1.5 s later.

(b) Calculate the vertical displacement of the rock 1.5 s later.

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Problem # H-5:

mechanics example prob kin h5

In order to calculate the height, h, of a building, a physics student launches a ball at an initial speed of 30 m/s at an angle of 65° above the horizontal. Another physics student is standing on the roof of the building, and determines that the ball lands on the roof of the building 3.5 s after it is launched.

(a) What is the height, h, of the building?

(b) What is the peak height, H, reached by the ball, relative to the student on the roof?

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Problem # H-6:

mechanics example prob kin h6

A person standing at the top of a hill throws a rock with an initial velocity of 14 m/s at an angle of 30° above the horizontal.

(a) Calculate the horizontal displacement of the rock 1.7 s later.

(b) Calculate the vertical displacement of the rock 1.7 s later.

(c) How long does it take the rock to fall 3.5 m below its initial launch height?

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Problem # H-7:

A ball is launched from the ground into the air. At a height of 7.3 m, the velocity of the ball is observed to be v = 8.2i + 5.7j in meters per second.

(a) What is the maximum height reached by the ball?

(b) What will be the total horizontal distance traveled by the ball?

(c) At the instant just before the ball hits the ground, what is the magnitude and direction of its velocity?

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Problem # H-8:

mechanics example prob kin h8

A stuntman rides his motorcycle at 25 m/s off a ramp that is inclined at 30°. He intends to land on the back of a truck. The driver of the truck must wait T seconds after the motorcycle launches off the ramp, before accelerating, from rest, at 5 m/s2, so that the motorcycle lands on the back of the truck. The edge of the ramp is 3 m high, the back of the truck is 1 m high, and the back of the truck is 45 m from the edge of the ramp, initially. What is the value of T ?

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Problem # H-9:

mechanics example prob kin h9

A baseball player throws a ball at an initial speed of 20 m/s, from point A, which is 1.4 m above the ground and 14 m from a wall. What is the launch angle θ so that the height h, which is the distance from the ground to the point of impact on the wall, is a maximum?

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Problem # H-10:

mechanics example prob kin h10

A ball is launched at ground level at a speed of 20 m/s, at an angle of 35° above the horizontal. A hill is located 25 m from the launch point, where it has an inclination of 20°. How far up the hill, D, does the ball land?

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Problem # H-11:

mechanics example prob kin h11

In a competition held in a high school gym, the goal is to launch a ball from floor level so that it passes through two rings suspended from the ceiling, as shown. One of the competitors is a physics student who calculates the value of L and θ based on a launch speed of 20 m/s. What are these two values?

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Here are additional projectile motion problems for you to work on.

Kinematics – Uniform circular motion – Senior high school and first year college/university

Problem # I-1:

A sprinter is running around the bend of a track with radius of 30 m, at a speed of 11 m/s. What is the acceleration of the sprinter and in what direction does the acceleration a point?

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Problem # I-2:

A charged particle moves in a circular path in a magnetic field, with a radial acceleration of 2.5 × 1014 m/s2. If the circular path has a radius of 14 cm, what is its speed?

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Problem # I-3:

A fan rotates at 1000 revolutions per minute. The tip of the blades have a radius of 0.20 m.

(a) What is the distance traveled by the tip of a blade during one full revolution?

(b) What is the speed of a blade tip?

(c) What is the acceleration of a blade tip?

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Problem # I-4:

A high speed train goes around a curve at a speed of 250 km/h. What is the smallest radius of curvature of the track so that the maximum acceleration experienced by the passengers is 0.05g ?

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Problem # I-5:

A high speed train goes around a curve with a radius of 1.5 km. What is the maximum speed of the train so that the maximum acceleration experienced by the passengers is 0.05g ?

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Problem # I-6:

mechanics example prob kin i6

A tether car is going around a track at a speed of 200 mph. To prevent the car from going off the track it is tethered to a center post with a cable. The diameter of the track is 21.3 m, and the wheel diameter of the car is 5 cm.

(a) What is the acceleration of the car?

(b) How fast do the car wheels rotate, in revolutions per minute?

(c) Looking at the car from above, how fast does it go around the track, in revolutions per minute?

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Kinematics – 1-D problems involving relative motion – Senior high school and first year college/university

Problem # J-1:

A boat is traveling in a river in the upstream direction, at 15 km/h with respect to the water of the river. The water in the river is flowing at 7 km/h with respect to the ground.

(a) What is the velocity of the boat with respect to ground?

(b) A person on the boat walks from the front of the boat to the back of the boat at 5 km/h with respect to the boat. What is the person's velocity with respect to the ground?

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Problem # J-2:

A person walks up an escalator that has stopped, in 100 seconds. When the escalator is moving, it takes the person 75 seconds to be carried up when they are standing on it. If the escalator is 20 m long, how long would it take that same person to walk up the moving escalator?

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Problem # J-3:

A cameraman is standing on the back of a pickup truck filming a scene for a movie. He videotapes a car traveling directly ahead of him moving at 45 mi/h faster than the truck. Suddenly, the car slows down, stops, and begins moving in the opposite direction at 60 mi/h, as measured by someone on the ground. If the pickup truck is moving at 35 mi/h, and the change in the car's velocity took 2.5 seconds, what was the acceleration of the car from the perspective of (a) the cameraman, and (b) the person on the ground?

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Kinematics – 2-D problems involving relative motion – Senior high school and first year college/university

Problem # K-1:

mechanics example prob kin k1

Two cars, A and B, are approaching an intersection as shown. Car A is 600 m from the intersection and is moving at 80 km/h. Car B is 700 m from the intersection and is moving at 90 km/h.

(a) In unit vector notation, what is the velocity of car A with respect to car B?

(b) How does the direction of the velocity found in (a) compare to the line of sight between the two cars?

(c) How does the answer to (a) and (b) change as the cars move closer to the intersection?

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Problem # K-2:

A train travels in the north direction at 25 m/s relative to the ground. At the same time it is raining. An observer on the ground sees that the raindrops make an angle of 18° with the vertical. A passenger on the train sees the raindrops fall in a perfectly vertical direction. What is the speed of the raindrops relative to the ground?

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Problem # K-3:

The pilot of a plane intends to fly directly east in the presence of a wind, a distance of 950 km. The plane has an airspeed of 630 km/h, and the pilot calculates that the plane must fly with a heading of 15° south of east. If the plane arrives at the destination 1.8 hours later, what was the velocity vector of the wind?

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Problem # K-4:

A 350 m wide river flows in the east direction at 2.5 m/s. A boat with a speed of 9.5 m/s relative to the water sets a course that is pointed in a direction 25° west of north.

(a) What is the velocity of the boat relative to the Earth?

(b) How long does it take the boat to cross the river, starting from the south bank?

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Problem # K-5:

A river flows in the east direction at 3.5 m/s. A boat with a speed of 8.5 m/s relative to the water sets a course that is pointed in a direction 30° north of east. Once the boat is in motion, one of the passengers walks from the left side of the boat directly to the right side of the boat at a speed of 1.5 m/s relative to the boat. What is the velocity of the passenger relative to the ground?

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Here are additional relative velocity problems for you to work on.



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