Question

A boy of mass \(60.0 \mathrm{kg}\) is rescued from a hotel fire by leaping into a firefighters' net. The window from which he leapt was \(8.0 \mathrm{m}\) above the net. The firefighters lower their arms as he lands in the net so that he is brought to a complete stop in a time of \(0.40 \mathrm{s}\). (a) What is his change in momentum during the 0.40 -s interval? (b) What is the impulse on the net due to the boy during the interval? [Hint: Do not ignore gravity.] (c) What is the average force on the net due to the boy during the interval?

Short answer

Question: Calculate the initial velocity of the boy just before he lands on the net, his change in momentum, the impulse on the net, and the average force on the net during his deceleration process. Answer: Using the provided information and the steps outlined above, we find that the initial velocity of the boy is approximately \(12.6 \mathrm{m/s}\). The change in momentum of the boy is approximately \(-756 \mathrm{kg\cdot m/s}\), which is the same as the impulse on the net due to the boy. The average force on the net due to the boy during the deceleration process is approximately \(-1890 \mathrm{N}\).

Step-by-step solution

Calculate the initial velocity of the boy

First, we need to find the initial velocity of the boy just before he lands on the net. Since the boy is in free fall, we can use the kinematic equation to find his initial velocity: \(v^2 = u^2 + 2as\) where \(v\) is the final velocity, \(u\) is the initial velocity (which is \(0\) as he starts from rest), \(a\) is the acceleration due to gravity (\(9.81 \mathrm{m/s^2}\)), and \(s\) is the vertical displacement (\(8.0 \mathrm{m}\)). After rearranging the equation, we get: \(v = \sqrt{2as}\)

Calculate the final velocity of the boy

Now, we need to find the final velocity of the boy after he comes to a complete stop. Since he stops in the net, his final velocity is \(0 \mathrm{m/s}\).

Calculate the change in momentum of the boy

To find the change in momentum of the boy, we can use the formula: \(\Delta p = m(v_f - v_i)\) where \(m\) is the mass of the boy (\(60.0 \mathrm{kg}\)), \(v_f\) is the final velocity (\(0 \mathrm{m/s}\)), and \(v_i\) is the initial velocity calculated in step 1. Plug the values into the formula to find the change in momentum.

Calculate the impulse on the net due to the boy

The impulse on the net is equal to the change in momentum of the boy. Since we found the change in momentum in step 3, we can use the same value for the impulse. This is because the impulse applied by the net should be equal and opposite to the momentum change of the boy.

Calculate the average force on the net due to the boy

To find the average force on the net during the interval, we can use the impulse-momentum theorem, which states that: \(I = Ft\) where \(I\) is the impulse, \(F\) is the average force, and \(t\) is the time interval (\(0.40 \mathrm{s}\)). Rearranging the formula, we get: \(F = \frac{I}{t}\) We can use the impulse calculated in step 4 and divide it by the given time interval to find the average force on the net due to the boy.