You have a arrived at the final accident scene of the day. Two cars of equal mass (2,000 kg each) Were involved in a head on accident at a 4 way stop intersection, here is what you know. ▪︎Car 1 started at rest and coasted toward the intersection from the top of a 50 meter hill.
▪︎Car 2 was on a flat stretch of road at the bottom of the hill in front of Car 1.
▪︎At the bottom of the hill, before breaking car 1 was going 30 m/s and car 2 was going 20 m/s.
▪︎From the skid marks on the road you can see that Car 1 applied force on its brakes for 5 seconds, 75 meters before this stop sign where it came to a stop.
▪︎There were no skid marks left by Car 2. The collision occurred at the stop sign, where Car 1 had stopped. After the collision, Car 2 came to a stop. Car 1's final velocity is unknown.
1. What was the potential energy of each car at the beginning of the problem?
2. What was the kinetic energy of each car while it was traveling at its maximum speed?
3. How much energy did Car 1 "lose" from the top to the bottom of the hill, before it began braking? What happened to the lost energy?
4. When Car 1 applied its brakes, how much force did its tires apply to the road to stop?
5. How much power (in watts) did it take to stop Car 1 by applying force to the brakes?
6. What is the momentum of each car right before the accident?
7. What was Car 1's final speed after the accident?

1) 980,000 J;  0 J

2) 900,000 J; 400,000 J

3) 80,000 J; converted into thermal energy

4) -12,000 N

5) 180,000 W

6) 60,000 kg m/s; 40,000 kg m/s

7) 20 m/s

Explanation:

1)

The potential energy of an object is the energy possessed by the object due to its location in the gravitational field; it is given by

where

m is the mass

is the acceleration due to gravity

h is the height above the ground

For car 1, when it is at the top of the hill,

m = 2000 kg

h = 50 m

So, its potential energy is

Car 2 instead was at the bottom of the hill, so h = 0, therefore its P.E. is zero:

PE = 0

2)

The kinetic energy of an object is the energy possessed by the object due to its motion. It is given by

where

m is the mass of the object

v is its speed

For car 1, at the bottom of the hill, we have

m = 2000 kg

v = 30 m/s

So its kinetic energy is:

For car 2, we have

m = 2000 kg

v = 20 m/s

So its kinetic energy is

3)

At the top of the hill, the total mechanical energy (sum of potential+kinetic energy) of car 1 was just equal to the initial potential energy:

At the bottom of the hill, the total mechanical energy is just equal to the final kinetic energy, so

Therefore, the energy lost is

The energy lost it has actually been converted into thermal energy, because of the presence of frictional forces that act against the motion of the car, slowing it down and converting part of its mechanical energy into kinetic energy.

4)

When car 1 applied force on its brakes, the work done on the car by the force is equal to its change in kinetic energy (work-energy theorem), so:

where:

F is the force applied

d = 75 m is the distance covered while  stopping

v = 0 is the final velocity (the car comes to a stop)

u = 30 m/s is the initial velocity when the car starts slowing down

m = 2000 kg is the mass of the car

Solving for F, we find:

where the negative sign indicates that the direction of the force is opposite to the direction of motion.

5)

The power used by car 1 to stop is given by

where:

is the change in energy of the car  while it stops

t is the time interval required for the car to stop

In this problem, we have:

is the magnitude of the kinetic energy lost by the car

t = 5 s is the time interval needed for the car to stop

Therefore, the power is

6)

The momentum of an object is given by

where

m is the mass of the object

v is its velocity

In this problem, for car 1 before the accident, we have

m = 2000 kg

v = 30 m/s

So, its momentum was

For car 2, before the accident we have

m = 2000 kg

v = 20 m/s

So, its momentum was

7)

Before the collision between the two cars:

- Car 1 has stopped, so its initial momentum is zero:

- Car 2 is moving at 20 m/s, so its initial momentum is

After the collision:

- Car 2 has stopped, so its final momentum is

- The final  velocity v of car 1 is unknown, so we can write its final momentum as

In a collision, the total momentum of the system is conserved, so we can write:

And by solving for v, we find the final velocity of car 1: