Imagine a car involved in a head-on crash. The driver, whose mass is m, is to be brought uniformly to rest within the passenger compartment by compressing an inflated air bag through a distance sc. Write an expression for the average force exerted on the bag in terms of m, vi, and sc. Compute the average force for a 100 km/h collision, where the driver’s mass is 60 kg and the allowed stopping distance in the bag is 30 cm.

To find the expression for the average force exerted on the airbag, we can use the principle of conservation of momentum. Assuming that the car and the driver were initially at rest, the momentum before the collision is zero. After the collision, the driver's momentum will also be zero since they are brought uniformly to rest.

The driver's initial momentum can be written as:

Pi = m * vi

where m represents the mass of the driver and vi is the initial velocity.

The final momentum of the driver is zero, so we can write:

Pf = m * 0 = 0

According to the principle of conservation of momentum, the change in momentum is equal to the impulse exerted on the driver. The impulse is given by the force multiplied by the time:

Impulse = F * t

Since the driver is brought to rest uniformly, we can relate the distance the airbag is compressed (sc) to the time (t) using the equation of motion:

sc = vi * t + (1/2) * a * t^2

where a is the acceleration of the driver, which is given by:

a = (vf - vi) / t

Since vf (final velocity) is zero, we have:

a = -vi / t

Rearranging the equation of motion to solve for t:

t = (sqrt(2*sc/a)) = (sqrt(2*sc*t^2/(−vi/t)))

Simplifying:

t = sqrt((2 * sc) / (-vi))

Now, substituting the expression for a in terms of t:

a = -vi / t = (-vi) / (sqrt((2*sc)/(-vi))) = -sqrt((-vi^2) / (2*sc))

Now that we have the time, we can compute the impulse:

Impulse = F * t = F * sqrt((2 * sc) / (-vi))

Since the impulse is the change in momentum, we can set it equal to Pi - Pf:

F * sqrt((2 * sc) / (-vi)) = Pi - Pf = m * vi - 0 = m * vi

Simplifying:

F = (m * vi) / sqrt((2 * sc) / (-vi))

Now, let's substitute the given values into the expression to find the average force for a 100 km/h collision.

Given:
vi = 100 km/h = 100000 m/3600 s = 27.78 m/s
m = 60 kg
sc = 30 cm = 0.3 m

Plugging the values into the expression:

F = (60 kg * 27.78 m/s) / sqrt((2 * 0.3 m) / (-27.78 m/s))

Calculating the average force will give you the final answer.