What was the speed of the Toyota at impact in this collision?

[camero33]

Need help!

A 1.0x10^3 kg Toyota collides into the rear end of a 2.2x10^3 kg Cadillac stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8m before stopping. The police officer, knowing that the coefficient of Kinectic friction between the tires and road is 0.40, calculates the speed of the Toyota at impact. What was that speed?

 

[Tide]

Exactly what have you tried so far?

HINT: Momentum is conserved!

 

[camero33]

ok i tried using this formula:
m1v1 + m2v2 = (m1 + m2) v(final)
m1v2 + m2v2 = (m1 + m2) times the square root of 2gd (gravity and distance)
then i isolated for v1 but i don't know how to apply the coefficient of friction to the problem...

 

[Tide]

I think you need to assume that the two cars instantly lock up and become a single object when the collision occurs (i.e. ignore the "crumple zone" and assume the vehicles cover no distance while becoming locked together.)

The combined system decelerates at the rate $\mu g$ (frictional force divided by mass) so you can relate the initial speed to the distance traveled before stopping with

$$V_0 = \sqrt {2 \mu g d}$$

You also know that

$$V_0 = \frac {m_1 v_1}{m_1+m_2}$$