Mechanics Exam - AP Physics C: Electricity and Magnetism
Card 1 of 1815
A car undergoes acceleration according to the given function. What distance has the car traveled after three seconds?
A car undergoes acceleration according to the given function. What distance has the car traveled after three seconds?
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We can determine the velocity by taking the second integral of acceleration for the time interval of 0s to 3s.
Solve for the first integral.
Solve for the second integral, using the time interval.
We can determine the velocity by taking the second integral of acceleration for the time interval of 0s to 3s.
Solve for the first integral.
Solve for the second integral, using the time interval.
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A car undergoes acceleration according to the given function. How fast is the car moving after four seconds?
A car undergoes acceleration according to the given function. How fast is the car moving after four seconds?
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We can find the car's velocity by taking the integral of the acceleration function during the given time interval.
Solve the integral for the time interval of 0s to 4s. This will give us the final velocity.
We can find the car's velocity by taking the integral of the acceleration function during the given time interval.
Solve the integral for the time interval of 0s to 4s. This will give us the final velocity.
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A car undergoes acceleration according to the given function. If the threshold for serious injury or fatality for a human undergoing horizontal acceleration is 60 g ees (1 gee = 10 meters per second per second), how long would a human be able to withstand riding in this car?
A car undergoes acceleration according to the given function. If the threshold for serious injury or fatality for a human undergoing horizontal acceleration is 60 g ees (1 gee = 10 meters per second per second), how long would a human be able to withstand riding in this car?
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Calculate the maximum acceration in meters per second.
Solve for the time.
Calculate the maximum acceration in meters per second.
Solve for the time.
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A ball travels with a velocity as described by the function below:
What is the ball's acceleration?
A ball travels with a velocity as described by the function below:
What is the ball's acceleration?
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Acceleration is equal to the derivative of the velocity function.
Since the velocity is constant, the derivative will be equal to zero.
The acceleration is equal to zero.
Acceleration is equal to the derivative of the velocity function.
Since the velocity is constant, the derivative will be equal to zero.
The acceleration is equal to zero.
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An object starts from rest and reaches a velocity of 
after accelerating at a constant rate for four seconds. What is the distance traveled in this time?
An object starts from rest and reaches a velocity of
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Since the acceleration is constant in this problem, we can apply the kinematics given equation to calculate the distance:
First, we need to calculate the acceleration.
Plug in our velocity and time values to find the acceleration.
Now we can return to the kinematics equation and solve for the distance traveled:
Since the acceleration is constant in this problem, we can apply the kinematics given equation to calculate the distance:
First, we need to calculate the acceleration.
Plug in our velocity and time values to find the acceleration.
Now we can return to the kinematics equation and solve for the distance traveled:
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At the moment a car is passed by another car constantly traveling at 
, it begins to accelerate at 
. In how many seconds does this car catch up to and pass the other car?
At the moment a car is passed by another car constantly traveling at
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First, find the displacement equations for both cars. Car 1 will be the car that is initially stationary; car 2 will be the car traveling with constant velocity.
Car 1:
Car 2:
Now, set those displacement equations equal to each other and solve for 
.
The accelerating car will catch and pass the car traveling at constant velocity after 4 seconds.
First, find the displacement equations for both cars. Car 1 will be the car that is initially stationary; car 2 will be the car traveling with constant velocity.
Car 1:
Car 2:
Now, set those displacement equations equal to each other and solve for
The accelerating car will catch and pass the car traveling at constant velocity after 4 seconds.
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An object moving along a line has a displacement equation of 
, where 
is in seconds. For what value of 
is the object stationary?
An object moving along a line has a displacement equation of
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Use the fact that 
to find the velocity equation, and then solve for 
when 
.
Take the derivative of the displacement equation.
Set the velocity equal to zero.
Solve for the time.
Use the fact that
Take the derivative of the displacement equation.
Set the velocity equal to zero.
Solve for the time.
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An object is moving on a line and has displacement equation of 
, where 
is in seconds. At what value of 
is the object not accelerating?
An object is moving on a line and has displacement equation of
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Take the second derivative of 
, to find the acceleration function 
. Then find the value of 
where 
. The fact that it is simply 
and not 
or 
inside that quantity means that using the Chain Rule becomes decidedly easier.
Set the acceleration fuction equal to zero and solve for the time.
Take the second derivative of
Set the acceleration fuction equal to zero and solve for the time.
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A projectile is launched out of a cannon or launch tube with zero air resistance or friction. At what angle (in degrees) should the projectile be launched to maximize the distance it travels?
A projectile is launched out of a cannon or launch tube with zero air resistance or friction. At what angle (in degrees) should the projectile be launched to maximize the distance it travels?
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The projectile travels the farthest when the vertical component of its velocity matches the sum of its horizontal component and whatever the wind/friction adds or subtracts. If the wind has no effect, then a 45-degree angle will be the best because the horizontal and vertical components of the velocity will create a right isosceles triangle (remember special triangles).
The projectile travels the farthest when the vertical component of its velocity matches the sum of its horizontal component and whatever the wind/friction adds or subtracts. If the wind has no effect, then a 45-degree angle will be the best because the horizontal and vertical components of the velocity will create a right isosceles triangle (remember special triangles).
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Which of the following is not a vector quantity?
Which of the following is not a vector quantity?
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Speed is defined as the magnitude of velocity.In other words, speed visually represents the size of a velocity vector, but is indepedent of its direction.
Velocity, acceleration, and force are all vector quantities. Each is defined by the magnitude of the measurement as well as its direction of action.
Speed is defined as the magnitude of velocity.In other words, speed visually represents the size of a velocity vector, but is indepedent of its direction.
Velocity, acceleration, and force are all vector quantities. Each is defined by the magnitude of the measurement as well as its direction of action.
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A dog is initially standing near a fire hydrant. The dog moves 3 meters to the right. Then it runs 7 meters to the left. What is the dog's final displacement from its original position?
A dog is initially standing near a fire hydrant. The dog moves 3 meters to the right. Then it runs 7 meters to the left. What is the dog's final displacement from its original position?
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This question tests your understanding of displacement as a vector quantity. The best way to solve it is to create a number line to track the motion of the dog.
Let the initial position of the dog be 0m on our number line:
Then the dog moves three meters to the right, which we can represent as follows:
Then the dog moves seven meters to the left. Note that it is moving seven meters to the left from the position it is at now (3m):
We find that at the end of its motion, the dog's displacement is four meters to the left of its original position, which we write as 
.
Remember that displacement is the change in position, and not the total distance traveled. We only care about the initial position and the final position, regardless of the path traveled. Positive and negative signs indicate the direction of motion.
This question tests your understanding of displacement as a vector quantity. The best way to solve it is to create a number line to track the motion of the dog.
Let the initial position of the dog be 0m on our number line:
Then the dog moves three meters to the right, which we can represent as follows:
Then the dog moves seven meters to the left. Note that it is moving seven meters to the left from the position it is at now (3m):
We find that at the end of its motion, the dog's displacement is four meters to the left of its original position, which we write as
Remember that displacement is the change in position, and not the total distance traveled. We only care about the initial position and the final position, regardless of the path traveled. Positive and negative signs indicate the direction of motion.
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You drive your car from your house all the way to your school which is 50km away. After you are done with classes you drive back through the same route and park exactly where you had your car at the beginning of the day. By the end of the day, what were the distance and displacement of your motion?
You drive your car from your house all the way to your school which is 50km away. After you are done with classes you drive back through the same route and park exactly where you had your car at the beginning of the day. By the end of the day, what were the distance and displacement of your motion?
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This question tests your conceptual understanding of distance as a scalar quantity vs your understanding of displacement as a vector quantity.
Distance measures the total length that was traveled in a given motion, and does not care about the direction since it is a scalar value. In your day, you traveled 50km on your way to school and 50km on your way back home. In total you traveled 100km, so that is your distance.
Displacement is a vector quantity that measures the change in position. It cares about your final and initial positions, taking into account the direction of the change in position. In this scenario you started and ended your motion exactly at the same position, so overall at the end of the day your car did not change position at all. Therefore your displacement was 0m.
This question tests your conceptual understanding of distance as a scalar quantity vs your understanding of displacement as a vector quantity.
Distance measures the total length that was traveled in a given motion, and does not care about the direction since it is a scalar value. In your day, you traveled 50km on your way to school and 50km on your way back home. In total you traveled 100km, so that is your distance.
Displacement is a vector quantity that measures the change in position. It cares about your final and initial positions, taking into account the direction of the change in position. In this scenario you started and ended your motion exactly at the same position, so overall at the end of the day your car did not change position at all. Therefore your displacement was 0m.
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An object travels North along a straight line at a constant rate of 
. What are the object's speed and velocity?
An object travels North along a straight line at a constant rate of
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This is a simple question that tests your conceptual understanding of speed as a scalar quantity and velocity as a vector quantity. The motion of the object is quite simple, so you need only to be mindful of the fact that velocity, as a vector, must tell you both the magnitude and direction (how fast it is going and where), while speed only tells you the magnitude (how fast it is going).
Therefore, your speed is 
and your velocity is 
.
This is a simple question that tests your conceptual understanding of speed as a scalar quantity and velocity as a vector quantity. The motion of the object is quite simple, so you need only to be mindful of the fact that velocity, as a vector, must tell you both the magnitude and direction (how fast it is going and where), while speed only tells you the magnitude (how fast it is going).
Therefore, your speed is
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The velocity (in meters per second) of a moving particle is given by the following function:
If the particle's initial position is 0m, what is the position of the particle after two seconds?
The velocity (in meters per second) of a moving particle is given by the following function:
If the particle's initial position is 0m, what is the position of the particle after two seconds?
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To solve this problem you need to obtain a function of position with respect to time. For that, you need to understand velocity as the rate of change of displacement with respect to time. In other words, velocity is "how fast" (i.e. how much time it takes) an object changes position (remember displacement is the change in position). This means that velocity is the derivative of displacement with respect to time.
Therefore, to obtain a function of position with respect to time you need to take the antiderivative of the velocity function, so we integrate:
Here, 
is a constant that represents the initial position of the particle. We know that the initial position of the particle is 0m, so our function is:
Therefore, after two seconds have passed, we have t = 2s and
Note: we know that position is given in meters since the question specified that velocity is measured in meters per second.
To solve this problem you need to obtain a function of position with respect to time. For that, you need to understand velocity as the rate of change of displacement with respect to time. In other words, velocity is "how fast" (i.e. how much time it takes) an object changes position (remember displacement is the change in position). This means that velocity is the derivative of displacement with respect to time.
Therefore, to obtain a function of position with respect to time you need to take the antiderivative of the velocity function, so we integrate:
Here,
Therefore, after two seconds have passed, we have t = 2s and
Note: we know that position is given in meters since the question specified that velocity is measured in meters per second.
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This morning you walked 20 meters to the right from your house to the bus stop, which took you 20 seconds. You waited at the bus stop for 1 minute before realizing you forgot your physics homework at home. You ran back to your house in 5 seconds. It took you 10 seconds to find your homework, and then you ran back to the bus stop in 5 seconds just in time to catch the bus. What was your average velocity for the entire period of motion?
This morning you walked 20 meters to the right from your house to the bus stop, which took you 20 seconds. You waited at the bus stop for 1 minute before realizing you forgot your physics homework at home. You ran back to your house in 5 seconds. It took you 10 seconds to find your homework, and then you ran back to the bus stop in 5 seconds just in time to catch the bus. What was your average velocity for the entire period of motion?
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This question tests your understanding of average velocity. You have different velocities throughout your motion: you changed direction and speed several times, including periods of no motion. However, average velocity cares only about your total displacement and how long it took you to get from your initial position to your final position.
Therefore, it is unnecessary to calculate your velocity at each time interval. You know your initial position is at your house and, after the entire motion, your final position is at the bus stop which is 20m to the right of your house. From this we get that the total displacement is 20m (note that it is positive because you moved to the right).
Since you moved back and forth and were motionless for some time intervals, it took you a total of 100s to finally get to the bus stop for good and catch the bus. You need to add all the times for each section of your motion; remember there are 60s in a minute and you need to be consistent with units.
Therefore, your average velocity is:
This question tests your understanding of average velocity. You have different velocities throughout your motion: you changed direction and speed several times, including periods of no motion. However, average velocity cares only about your total displacement and how long it took you to get from your initial position to your final position.
Therefore, it is unnecessary to calculate your velocity at each time interval. You know your initial position is at your house and, after the entire motion, your final position is at the bus stop which is 20m to the right of your house. From this we get that the total displacement is 20m (note that it is positive because you moved to the right).
Since you moved back and forth and were motionless for some time intervals, it took you a total of 100s to finally get to the bus stop for good and catch the bus. You need to add all the times for each section of your motion; remember there are 60s in a minute and you need to be consistent with units.
Therefore, your average velocity is:
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A block of mass 
is attached to two springs, each of whose spring constant is 
. The ends of the springs are fixed, and the block is free to move back and forth. It is released from rest at an initial amplitude, and its period is measured to be 
. What would the period be if the spring on the right side were to be moved to the other side, attached along side of the other spring?
A block of mass
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Because the springs are effectively in a parallel arrangement already, moving one does not change the effective spring constant, and therefore does not affect the period.
Because the springs are effectively in a parallel arrangement already, moving one does not change the effective spring constant, and therefore does not affect the period.
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A 5.0 kg mass oscillates once. The total distance it travels is 1.5 m and it takes 4.0 s to travel that distance. What is its frequency of oscillation?
A 5.0 kg mass oscillates once. The total distance it travels is 1.5 m and it takes 4.0 s to travel that distance. What is its frequency of oscillation?
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Frequency is only based on the period of the oscillation; all the other given information is useless for this problem. Using 
, we can calculate that the frequency is 0.25 Hz.
Frequency is only based on the period of the oscillation; all the other given information is useless for this problem. Using
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A mass is attached to a spring, which is fixed to a wall. The mass is pulled away from the spring's equilibrium point and is then released. At what point does the mass experience its maximum kinetic energy?
A mass is attached to a spring, which is fixed to a wall. The mass is pulled away from the spring's equilibrium point and is then released. At what point does the mass experience its maximum kinetic energy?
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The formula for determining kinetic energy is
So, kinetic energy will be greatest when the mass is moving most quickly. The force of the spring on the mass increases the mass's velocity until the spring’s equilibrium point, where the force of the spring acts against the motion of the mass, slowing it down. The mass is moving fastest at the spring's equilibrium point, so that's where its kinetic energy is greatest.
The formula for determining kinetic energy is
So, kinetic energy will be greatest when the mass is moving most quickly. The force of the spring on the mass increases the mass's velocity until the spring’s equilibrium point, where the force of the spring acts against the motion of the mass, slowing it down. The mass is moving fastest at the spring's equilibrium point, so that's where its kinetic energy is greatest.
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A mass oscillates on a spring with period 
. If the mass is doubled, what is the new period of oscillation?
A mass oscillates on a spring with period
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The formula for the period of oscillation is
.
When we double the mass, we get:
Because the new factor of 2 is under the square root sign, and also in the numerator, the new period will be increased by 
.
The formula for the period of oscillation is
When we double the mass, we get:
Because the new factor of 2 is under the square root sign, and also in the numerator, the new period will be increased by
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A mass is attached to a spring with a spring constant of 
. The mass is moved 2 m from the spring's equilibrium point. What is the total energy of the system?
A mass is attached to a spring with a spring constant of
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The total energy of the system is
but due to the face that the mass currently has no velocity, the kinetic energy term goes to zero.
Plugging in the given values, we can solve for the total energy of the system:
The total energy of the system is
but due to the face that the mass currently has no velocity, the kinetic energy term goes to zero.
Plugging in the given values, we can solve for the total energy of the system:
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