In a simulated head-on crash test, a car impacts a wall at 25 mi/h (40 km/h) and comes abruptly to rest. A 120 lb passenger dummy (with a mass of 55 kg), without a seat belt, is stopped by an air bag, which exerts a force on the dummy of 3000 lb. How long was the dummy in contact with the air bag while coming to a stop?
To calculate the time the dummy was in contact with the airbag while coming to a stop, we can use the principles of physics and the concept of impulse.
Impulse is defined as the product of force and time, and it is equal to the change in momentum of an object. Mathematically, impulse (J) can be represented as:
J = F * Δt
Where:
J = Impulse
F = Force exerted by the airbag
Δt = Change in time
We can rearrange this equation to solve for time (Δt):
Δt = J / F
Given that the force exerted by the airbag is 3000 lb (lb stands for pounds), we need to convert the units to a consistent system. Let's convert it to Newtons (N) because SI units are typically used in physics calculations:
1 lb ≈ 4.448 N
Therefore, 3000 lb (force) ≈ 3000 * 4.448 N
Now, the force exerted by the airbag is 13344 N.
We know the force of the impact but not the time of impact (Δt). However, we can calculate the change in momentum (Δp) of the dummy using the following formula:
Δp = m * Δv
Where:
Δp = Change in momentum
m = Mass of the dummy
Δv = Change in velocity
To calculate Δv, we need to convert the initial velocity of the car to m/s since SI units are required. We know that 1 mi/h (mile per hour) is approximately 0.4470 m/s.
Given that the car impacts the wall at 25 mi/h, the initial velocity can be calculated as:
Initial velocity (v₁) = 25 mi/h * 0.4470 m/s
Now, we can use Δv to calculate the change in momentum:
Δv = (0 - v₁) [Because the car comes abruptly to rest, its final velocity is 0]
Finally, we can substitute the values into the equation to solve for Δp:
Δp = m * Δv
Given that the mass of the dummy is 55 kg, we can calculate Δp.
Once we have calculated Δp, we can substitute the values of Δp and F into the equation to determine the time:
Δt = J / F
Δt = Δp / F
The calculated value of Δt will give us the duration of contact between the dummy and the airbag while coming to a stop. Let's calculate it!