The easiest way to approach this problem is find the average velocity, multiply by 2 because the car started from rest, and then divide the final velocity by time to get the acceleration. 

Average velocity:

Final velocity:

The final velocity is   because the car started from rest. This is evident from the equation for average velocity if we solve for final velocity:

Acceleration:

Knowns:

Unknowns:

 that Trevor actually travels = ?

Equation:

Since Trevor takes a moment to react before stepping on the breaks, determine how far Trevor travels during the reaction time. At this time he is traveling at a constant velocity.

Next, using kinematic equations to determine where Trevor stops his car based on the acceleration of his car.

 beyond the reaction point

Total distance traveled is the reaction time distance plus the distance traveled during the deceleration period.

total distance traveled 

To find the distance beyond the red light, subtract the distance traveled from the distance to the light

distance beyond red light 

Knowns:

Unknowns:

Equation:

To make things easier, set the frame of reference to the police officer.  This means that the police officer is traveling at 0km/h and the speeder is moving at a speed relative to the officer.

So the new relative velocities are:

This will help make the math easier.

Convert the relative velocity of the speeder to m/s.

Now there are two equations that have to be true for the officer to catch up to the speeder.

There are two things that connect these equations.  First, the distance that the officer travels and the speeder travels must be the same.  Additionally the final velocity of both the officer and the speeder must be the same.

Rearrange the speeder equation for the distance traveled.

Set the two equations equal, as the distance traveled of the speeder is equal to the distance traveled by the officer.

Remember that the reference point is at the officer’s perspective so the officer has an initial velocity of 0m/s.

Rewrite the equation breaking up the change of time into time final and time initial.

Distribute on the left side.

Substitute values

Simplify

Remember that the final time for both the officer and the speeder are the same.  So rearrange this in the form of a quadratic equation.

Use the quadratic formula to solve.  

Possible answers:  or 

Since the officer did not move until , then the second answer must be correct.

We are given final velocity and we know initial velocity is 0. We can use the following kinematics formula to relate final speed, initial speed, acceleration, and displacement:

The initial velocity is zero since Usain starts from rest. Therefore we can remove it from the equation.

Rearrange the equation to solve for acceleration.

Knowns:

Unknowns:

Equation:

First we need to determine Peter’s acceleration. The best kinematic equation to use here is:

Once Peter’s acceleration is known it is possible to find the distance he traveled.

Peter’s initial velocity is  so this equation simplifies to

The distance traveled is directly related to the square of the time traveled.  Therefore time if time is doubled, the value will be squared and therefore four times as great.  The distance traveled will be 4 times as far.

The can will have the same acceleration both up and down the hill since there is no friction.  Friction or other forces would cause a change in acceleration.  But since there is no friction, the can will travel up the hill, slow down and then accelerate back down the hill.

Knowns:

Unknown:

Equation:

The first thing is to determine the initial velocity of the runner before the runner accelerates for the final portion of the race.  Since the runner is traveling at a constant velocity

Next, convert the time during the constant speed portion to seconds.

Determine the amount of distance traveled while at a constant speed.

Use these values to determine the velocity of the runner.

Next determine the final velocity needed for the runner to finish out the race in the remaining time.

 left in race

Finally use the kinematic equations to calculate the time needed to get to this velocity.

Rearrange for time

Explanation:

We are given the initial velocity, time, and final velocity (zero because the ball stops). Using these values and the appropriate motion equation, we can solve for the acceleration.

Acceleration is given by the change in velocity over time:

We can use our values to solve for the acceleration.

There is a direct relationship between the final velocity and the time traveled.  Therefore if the time is doubled, the velocity would double as well.