A 52.0 kg male dancer leaps 0.37 m high.
(a) With what momentum does he reach the ground?
(c) As the dancer lands, his knees bend, lengthening the stopping time to 0.050 s. Find the average force exerted on the dancer's body.
(d) Compare the stopping force with his weight.
(stopping force/dancer's weight)
what is the time first of all and then what do i do after that??
In this problem, you are given the mass of the dancer (52.0 kg) and the height of his leap (0.37 m). You are asked to find the momentum with which he reaches the ground, the average force exerted on his body as his knees bend, and the ratio of the stopping force to his weight.
(a) To find the momentum with which the dancer reaches the ground, you can use the principle of conservation of mechanical energy. At the highest point of the leap, the dancer will have only potential energy, given by the equation:
PE = mgh
where PE is the potential energy, m is the mass of the dancer, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the leap.
In this case, the potential energy at the highest point of the leap is converted into kinetic energy just before the dancer reaches the ground. The kinetic energy is given by the equation:
KE = (1/2)mv^2
where KE is the kinetic energy and v is the velocity of the dancer just before reaching the ground.
Since the kinetic energy and potential energy are equal, we can equate the two equations:
PE = KE
mgh = (1/2)mv^2
Simplifying and solving for v:
v^2 = 2gh
v = sqrt(2gh)
Substituting the given values into the equation (m = 52.0 kg, g = 9.8 m/s^2, and h = 0.37 m):
v = sqrt(2 * 9.8 m/s^2 * 0.37 m)
v ≈ 3.41 m/s
To find the momentum, you can multiply the mass of the dancer with the velocity just before reaching the ground:
momentum = m * v
momentum = 52.0 kg * 3.41 m/s
momentum ≈ 177 kg·m/s
So, the momentum with which the dancer reaches the ground is approximately 177 kg·m/s.
(c) To find the average force exerted on the dancer's body as his knees bend, you can use the impulse-momentum theorem. The impulse experienced during the bending of the knees is equal to the change in momentum. The impulse can be calculated using the equation:
impulse = change in momentum
impulse = m * Δv
where impulse is the average force multiplied by the time interval Δt, m is the mass of the dancer, and Δv is the change in velocity (from v to 0 m/s) as the dancer comes to a stop.
Given that Δt = 0.050 s and Δv = -3.41 m/s (negative sign because the dancer is coming to a stop), we can calculate the impulse:
impulse = m * Δv
impulse = 52.0 kg * (-3.41 m/s)
impulse ≈ -177 kg·m/s
The negative sign indicates that the impulse is in the opposite direction to the motion of the dancer.
Since impulse is equal to the average force multiplied by the time interval, we can rearrange the equation to find the average force:
average force = impulse / Δt
average force = -177 kg·m/s / 0.050 s
average force ≈ -3540 N
So, the average force exerted on the dancer's body as his knees bend is approximately -3540 N.
(d) To compare the stopping force with the dancer's weight, you can calculate the ratio of the stopping force to his weight:
stopping force / dancer's weight = average force / (m * g)
Substituting the given values:
stopping force / dancer's weight = -3540 N / (52.0 kg * 9.8 m/s^2)
stopping force / dancer's weight ≈ -7.08
The negative sign indicates that the stopping force is in the opposite direction to the dancer's weight.
So, the ratio of the stopping force to the dancer's weight is approximately -7.08.