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?
The 3000 lb force exerted by the air bag is 13,360 Newtons.
Force x time = Momentum change = M V
The initiaL dummy momentum is 55 kg*11.12 m/s = 612 kg m/s
time = 612 kg m/s/13,360 kg m/s^2 = 0.046 s
To find the duration of contact with the airbag, we can use the impulse-momentum principle. According to this principle, the change in momentum of an object is equal to the impulse applied to the object. The impulse can be calculated using the equation:
Impulse = Force × Time
In this case, the impulse is equal to the change in momentum of the passenger dummy, which we can calculate using the equation:
Impulse = Mass × Change in Velocity
The initial velocity of the dummy is 40 km/h (which needs to be converted to m/s) and the final velocity is 0 m/s since the dummy comes to rest. We can calculate the change in velocity by subtracting the final velocity from the initial velocity.
Let's break down the steps to find the duration of contact:
Step 1: Convert the initial velocity of the dummy from km/h to m/s.
To convert from km/h to m/s, we use the conversion factor: 1 km/h = 0.2778 m/s.
Therefore, the initial velocity (v) of the dummy is: v = 40 km/h x 0.2778 m/s = 11.112 m/s.
Step 2: Calculate the change in velocity.
The change in velocity (Δv) is equal to the final velocity (0 m/s) minus the initial velocity (11.112 m/s).
Δv = 0 m/s - 11.112 m/s = -11.112 m/s.
Note that the negative sign indicates a decrease in velocity.
Step 3: Calculate the impulse.
Using the equation Impulse = Mass × Change in Velocity, we can substitute the values:
Impulse = (55 kg) × (-11.112 m/s) = -610.16 kg·m/s.
Note that the impulse is negative because it is in the opposite direction of the initial velocity.
Step 4: Calculate the duration of contact.
Using the equation Impulse = Force × Time, we can rearrange it to solve for time:
Time = Impulse / Force.
Time = (-610.16 kg·m/s) / (3000 lb).
However, we need to convert the force to SI units before plugging it in. To convert pounds (lb) to Newtons (N), we use the conversion factor: 1 lb = 4.44822 N.
Therefore, the force (F) applied by the airbag is: F = 3000 lb x 4.44822 N/lb.
Now let's calculate the duration of contact:
Force = 3000 lb x 4.44822 N/lb = 13344.66 N
Time = (-610.16 kg·m/s) / 13344.66 N
Using a calculator, you can divide the impulse by the force to obtain the duration of contact in seconds.
To solve this problem, we can use Newton's second law of motion which states that force is equal to mass multiplied by acceleration (F = ma) and the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
First, let's convert the speed of the car from miles per hour (mi/h) to meters per second (m/s):
25 mi/h * (1.60934 km/mi) * (1000 m/km) * (1 hour/3600 s) = 11.176 m/s
Next, let's convert the force exerted by the airbag from pounds (lb) to newtons (N):
3000 lb * (4.44822 N/lb) = 13344.66 N
Now, let's calculate the acceleration of the dummy using Newton's second law:
F = ma
13344.66 N = 55 kg * a
a = 13344.66 N / 55 kg
a ≈ 242.63 m/s²
Since the final velocity (v) is 0 m/s (the dummy comes to a stop), and the initial velocity (u) is 11.176 m/s, we can use the equation v = u + at to calculate the time (t) it takes for the dummy to come to a stop:
0 = 11.176 m/s + 242.63 m/s² * t
-11.176 m/s = 242.63 m/s² * t
t = -11.176 m/s / 242.63 m/s²
t ≈ -0.046 s
Since time cannot be negative, we round the value to the nearest positive value. Therefore, the dummy was in contact with the airbag for approximately 0.046 seconds.