Crash and Whiplash, Part Deux

Classical Mechanics Level pending

This problem explores one of the assumptions of this problem . To recapitulate, a truck of mass 20 tons collides head on with a car of 1.5 tons. Both are moving at 15 m/s. The earlier problem assumed the period of the collision was 100 milliseconds. If the crumple zone of the vehicle is 1 meter, how long does the collision last, from the moment of impact until both car and truck are moving together with the same velocity?

Assumptions:

  • The collision is perfectly inelastic

  • The car crumples uniformly, with a constant impulse force over the period of the collision

  • The truck has no crumple zone

The answer is in milliseconds, rounded to the nearest integer.

31 72 100 17 36 62

G Silb by G Silb , 56, USA

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1 solution

G Silb
Aug 15, 2019

Let M M designate the mass of the truck, m m the mass of the car, v v their velocity before impact, and V V the velocity after.

Since momentum is conserved and the collision is inelastic: m ( − v ) + M v = ( m + M ) V m (-v) + M v = (m+M) V

Then the change in velocity experienced by the car is Δ v = m ( − v ) + M v m + M + v = 2 M v M + m \Delta v = \frac{m(-v) + M v}{m+M} + v = \frac{2 M v}{M+m}

The distance traveled during this change in velocity is 1 2 Δ v t = d \frac{1}{2}\ \Delta v t = d

Solving, t = 2 d / Δ v = d ( m + M ) M v t = 2 d / \Delta v = \frac{d(m+M)}{M v}

Substituting in the numbers, t = 1 ∗ ( 1.5 + 20 ) 20 ∗ 15 ≈ 72 t= \frac{1 *(1.5+20)}{20*15} \approx 72

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