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EXAMPLE: COEFFICIENT OF RESTITUTION

PROBLEM STATEMENT

A speeding car (1500 kg) strikes a stopped van (2500 kg). A post-accident forensic investigation determines that the van was imparted with a speed of 15 m/s immediately after the impact. Previous investigators have determined that collisions of this nature tend to have a coefficient of restitution of about 0.5. Estimate the speed that the car was traveling at when it struck the van.

SOLUTION METHOD

This problem can be solved by applying conservation of momentum and the coefficient of restitution.

m₁v_1, i + m₂v_2, i = m₁v_1, f + m₂v_2, f

e = (v_2, final − v_1, final)/(v_2, initial − v_1, initial)

Thus we can express this as two algerbraic equations with two unknowns: v_1, i and v_1, f.

1500v_1, i + 0 = 1500v_1, f + (2500)(15)

0.5 = (15 − v_1, f)/(0 − v_1, i)

Writing this in matrix form:

1500 − 1500 0.5 − 1 v_1, i v_1, f = (2500)(15) − 15

Solving this, we find that the initial speed of the vehicle was about 80 m/s or nearly 180 mph! We can also find that the car’s final velocity was 55 m/s.