Average force of a head in a collision

Question

Car manufacturers conduct crash tests on their cars in order to improve crash safety. In the event of a crash the head of any child travelling in the front seat can strike the glove compartment at a considerable (relative) speed, even if the child is wearing a seat belt.

a) The manufactures of a particular brand of car conduct head on collision tests and find that in the absence of a passenger side airbag, a child's head which has a mass of 3.5 kg, goes from a speed of 40 km h^-1 relative to the dashboard just before its collision to rebounding from the dashboard at 15 km h^-1 just after the collision. This collision lasts just 0.08 seconds. What is the average force exerted on the child's head during the crash?

b) The manufacturer wishes to reduce the average force involved in such a collision to 200 N. In order to achieve this they install a passenger airbag on the front of the glove compartment which quickly inflates in the event of a crash and deflates as the child's head pushes into it, effectively increasing the amount of time it takes to sloe the child's head. How long would the collision between the child's head and the airbag need to last to reduce the speed of the head relative to the dashboard from 40 km h^-1 without exceeding the average force quoted above?

Answer 1

Car manufacturers conduct crash tests on their cars in order to improve crash safety. In the event of a crash the head of any child travelling in the front seat can strike the glove compartment at a considerable (relative) speed, even if the child is wearing a seat belt.

a) The manufactures of a particular brand of car conduct head on collision tests and find that in the absence of a passenger side airbag, a child's head which has a mass of 3.5 kg, goes from a speed of 40 km h^-1 relative to the dashboard just before its collision to rebounding from the dashboard at 15 km h^-1 just after the collision. This collision lasts just 0.08 seconds. What is the average force exerted on the child's head during the crash?

FΔt=m(v₂-v₁)

F=m|(v₂-v₁)|/Δt=

(3.5 kg)|(15-40) (km/hr)(1000m/km)(hr/60² sec|/0.08 sec=303.82 N

Solution: 0.30 kN

b) The manufacturer wishes to reduce the average force involved in such a collision to 200 N. In order to achieve this they install a passenger airbag on the front of the glove compartment which quickly inflates in the event of a crash and deflates as the child's head pushes into it, effectively increasing the amount of time it takes to sloe the child's head. How long would the collision between the child's head and the airbag need to last to reduce the speed of the head relative to the dashboard from 40 km h^-1 without exceeding the average force quoted above?

If F=200 N,

F=m|v₂-v₁|/Δt

Δt=m|v₂-v₁|/F=(3.5 kg)|15-40|(1000/60²)/200=0.12 sec

Solution: 0.12 sec