A 1000 kg Toyota, initially traveling at 40 mph, collides into the rear end of a 2000 kg Cadillac, initially stopped at a red light. The bumpers lock and the two cars skid for 2 seconds before coming to rest. Calculate the coefficient of kinetic friction between the tires and the road that is needed to bring the two cars to rest in 2 seconds.

find the velocity after collision

1000*v=(1000+2000)V
v= convert 40mph to m/s
solve for V.
KE after collision= 1/2 (1000+2000)V^2
friction work: mu*(totalmass)g*distance

but distance=avg velocity*time
= 1/2 V*t, t=2
solve for mu.

To calculate the coefficient of kinetic friction between the tires and the road, we can use Newton's second law of motion and the concept of work done.

1. Start by converting the given speeds from mph to m/s:
- The initial speed of the Toyota, 40 mph, can be converted to m/s by multiplying it by 0.44704 (1 mph = 0.44704 m/s).
So, the initial speed of the Toyota is 40 mph * 0.44704 m/s/mph = 17.882 m/s.

2. Next, use the concept of conservation of momentum:
- The total momentum before the collision is given by the sum of the individual momenta of the two cars.
Momentum (p) = mass (m) * velocity (v).
The momentum of the Toyota before the collision is 1000 kg * 17.882 m/s = 17882 kg·m/s.
The momentum of the Cadillac is 0 kg·m/s since it is initially at rest.

3. During the collision, the two cars skid for 2 seconds before coming to rest. We need to find the acceleration (a) during this time:
- Using the first equation of motion: v = u + at, where v is final velocity, u is initial velocity, a is acceleration, and t is time.
Since the cars come to rest, the final velocity is 0 m/s, the initial velocity for the Toyota is 17.882 m/s, and the time is 2 seconds.
Therefore, 0 m/s = 17.882 m/s + a * 2 s.
Rearranging the equation gives: a = -17.882 m/s / 2 s = -8.941 m/s².
Note that the negative sign indicates that the acceleration is in the opposite direction of the initial velocity.

4. Now we can calculate the net force acting on the system by using Newton's second law: F = ma.
- The net force acting on the system is equal to the force of friction:
F = µ * m * g, where µ is the coefficient of kinetic friction, m is the mass of the Toyota, and g is the acceleration due to gravity (approximately 9.8 m/s²).
Rearranging the equation gives: µ = F / (m * g).

5. To find the force of friction, we need to calculate the change in momentum of the Toyota during the collision:
- Using the formula: ∆p = F * t, where ∆p is the change in momentum during the collision, F is force, and t is time.
Since the initial momentum is 17882 kg·m/s and the final momentum is 0 kg·m/s, the change in momentum is -17882 kg·m/s.
Rearranging the equation gives: F = ∆p / t = -17882 kg·m/s / 2 s = -8941 kg·m/s².

6. Now substitute the calculated values back into the equation from step 4 to calculate the coefficient of kinetic friction:
- µ = (-8941 kg·m/s²) / (1000 kg * 9.8 m/s²) = -0.914.

Since coefficients of friction are always positive, we can take the absolute value of -0.914, resulting in a coefficient of kinetic friction of approximately 0.914.