A boy of mass 64.1 kg is rescued from a hotel fire by leaping into a firefighters' net. The window from which he leapt was 7 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 s. Ignore air resistance.
(a) What is his change in momentum during this time 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?
To find the answers to these questions, we need to use the principles of momentum and impulse.
(a) The change in momentum of an object can be calculated using the formula:
Change in momentum = mass × velocity
Since the boy is brought to a complete stop, his final velocity is zero. Therefore, the change in momentum is equal to the initial momentum. The initial momentum of the boy can be calculated using the formula:
Initial momentum = mass × initial velocity
The initial velocity can be found using the formula for gravitational potential energy:
Potential energy = mass × gravity × height
where gravity is approximately 9.8 m/s^2.
Substituting the values into the equation, we have:
Potential energy = 64.1 kg × 9.8 m/s^2 × 7 m
Once you have the potential energy, you can find the initial velocity using the formula:
Potential energy = 1/2 × mass × initial velocity^2
Solving for initial velocity:
Initial velocity = √(2 × potential energy / mass)
Finally, the change in momentum is the initial momentum, which is equal to the mass times the initial velocity.
(b) The impulse on an object can be calculated using the formula:
Impulse = change in momentum
So, the impulse on the net is equal to the change in momentum of the boy.
(c) The average force on an object can be calculated using the formula:
Average force = impulse / time
So, the average force on the net due to the boy is equal to the impulse divided by the time interval.