A 1340-kg car traveling
east at 13.6 m/s (20 mi/h) has a head-on collision with a
1930-kg car traveling west at 20.5 m/s (30 mi/h). If the collision
time is 0.10 s, what is the force needed to restrain a 68-kg
person in the smaller car? In the larger car?
To calculate the force needed to restrain a person in each car during the head-on collision, we need to apply the principles of Newton's second law of motion and consider the conservation of momentum.
Let's break down the problem into smaller steps:
Step 1: Convert the speeds of the cars from mph to m/s.
Given:
Speed of the smaller car = 13.6 m/s
Speed of the larger car = 20.5 m/s
Step 2: Calculate the change in velocity of each car.
The change in velocity can be found by subtracting the final velocity from the initial velocity for each car.
Given:
Change in velocity of the smaller car = Speed of the larger car - Speed of the smaller car
= 20.5 m/s - 13.6 m/s
= 6.9 m/s (towards the east)
Change in velocity of the larger car = Speed of the smaller car - Speed of the larger car
= 13.6 m/s - 20.5 m/s
= -6.9 m/s (towards the west)
Step 3: Calculate the momentum of each car before the collision.
Momentum is calculated by multiplying the mass of an object by its velocity.
Given:
Mass of the smaller car = 1340 kg
Mass of the larger car = 1930 kg
Momentum of the smaller car = Mass of the smaller car * Speed of the smaller car
Momentum of the larger car = Mass of the larger car * Speed of the larger car
Step 4: Calculate the change in momentum of each car.
The change in momentum can be found by subtracting the initial momentum from the final momentum for each car.
Given:
Change in momentum of the smaller car = Momentum of the larger car - Momentum of the smaller car
Change in momentum of the larger car = Momentum of the smaller car - Momentum of the larger car
Step 5: Calculate the average force required to change the momentum of each car.
Using Newton's second law of motion, the average force can be calculated by dividing the change in momentum by the collision time.
Given:
Collision time = 0.10 s
Average force on the smaller car = Change in momentum of the smaller car / Collision time
Average force on the larger car = Change in momentum of the larger car / Collision time
Step 6: Calculate the force needed to restrain a 68-kg person in each car.
Given:
Mass of person = 68 kg
Force needed to restrain a person in the smaller car = Average force on the smaller car
Force needed to restrain a person in the larger car = Average force on the larger car
By following these steps and plugging in the given values, you should be able to calculate the force needed to restrain a person in each car during the collision.