# Example Mechanics Problems

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 – 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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

Problem # A-2: 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?

Problem # A-3: Given the above graph of a(t), sketch the graph of v(t) and d(t).

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?

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.

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?

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?

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?

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?

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?

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?

Problem # B-3: 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?

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)).

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 ?

Problem # B-6: 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 ?

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?

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 ?

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.

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 ?

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?

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 ?