A man claims he can safely hold on to a 14.20 kg child in a head-on collision with a relative speed of 132. mi/h lasting for 0.100 s as long as he has his seat belt on.
(a) Find the magnitude of the average force needed to hold onto the child.
To find the magnitude of the average force needed to hold onto the child, we can use the impulse-momentum principle. According to this principle, the impulse experienced by an object is equal to the change in its momentum. The impulse is given by the product of the average force applied to the object and the time during which the force is applied.
The formula to calculate impulse is given by:
Impulse = Force × Time
We are given the following information:
Mass of the child, m = 14.20 kg
Relative speed = 132 mi/h
Duration of the collision, t = 0.100 s
First, we need to convert the relative speed from miles per hour to meters per second:
Relative speed = 132 mi/h * (1 h / 3600 s) * (1609.34 m / 1 mi) = 59.08 m/s
Next, we can calculate the initial momentum of the child using the formula:
Initial momentum = Mass × Initial velocity
The initial velocity of the child is the relative speed, so:
Initial momentum = m × relative speed
Now we can calculate the initial momentum:
Initial momentum = 14.20 kg × 59.08 m/s = 839.576 kg·m/s
Since the child comes to rest after the collision, the final momentum is given by:
Final momentum = 0 (as the child has zero velocity after the collision)
Using the impulse-momentum principle, we have:
Impulse = Final momentum - Initial momentum
Rearranging the equation to solve for force:
Force = Impulse / Time
Substituting the values:
Force = (0 - 839.576 kg·m/s) / 0.100 s
Calculating the force:
Force = -8395.76 N (note that the negative sign indicates that the force is in the opposite direction to the initial momentum)
Finally, the magnitude of the average force needed to hold onto the child is:
Magnitude of the force = |-8395.76 N| = 8395.76 N (since force is a scalar value)
Therefore, the magnitude of the average force needed to hold onto the child is approximately 8395.76 Newtons.