A person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of . Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the floor? (f) Does it make sense to say that this energy came from the floor? Explain.
(a) Yes, the impulse is imparted to the person by the floor.
(b) Yes, the floor does a certain amount of work on the person.
(c) The momentum when the person leaves the floor is .
(d) Yes, it can be said that the momentum came from the floor.
(e) The kinetic energy when the person leaves the floor is .
(f) The kinetic energy required for the person leave the floor does not come from the floor.
The mass of the person is, .
The maximum height of the center of mass of the person is, .
When a body moves to a distance by applying an external force on it, then the work done is equal to the total change in the mechanical energy of the body.
If the body is moving vertically, then the height up to which the body moves relies upon the body’s initial velocity when it leaves the ground.
When the person jumps straight up from the ground, then the weight of the person exerts force on the ground and equal but opposite amount of force is exerted by the ground on the person’s feet. It means that the momentum of the person changes from before the jump to after the jump.
The change in the momentum of the person produces the impulse to the person.
Hence, yes, the impulse is imparted to the person by the floor.
The force is exerted on the person by the ground and the person jumps to a certain height. The work is done on the person by the ground to reach the maximum height.
Hence, yes, the floor does a certain amount of work on the person.
Applying conservation of energy to the system,
At maximum height, the final velocity of the person is 0 and at the initial position the person is on the ground. So, putting and in the equation,
The value of the momentum when the person leaves the floor is given by,
Hence, the momentum when the person leaves the floor is .
It is given that the person exerts a certain amount of force as well as the momentum on the floor. Then, based on the conservation of momentum, the floor exerts the same magnitude of momentum but in opposite direction to the person.
Hence, yes, it can be said that the momentum came from the floor.
The formula for the kinetic energy when the person leaves the floor is given by,
Hence, the kinetic energy when the person leaves the floor is .
When the person bends his knees and then jumps straight up, the elastic potential energy stored in the muscles is converted into the kinetic energy required for jumping.
It means that the kinetic energy does not come from the floor, it comes from the elastic potential energy stored in the muscles of the person.
Hence, the kinetic energy required for the person leave the floor does not come from the floor.
In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person’s heart in one stroke. This information can be obtained by means of a ball is to cardiograph. The instrument works as follows. The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass m of blood into the aorta with speed v, and the body and platform move in the opposite direction with speed V. The blood velocity can be determined independently (e.g., by observing the Doppler shift of ultrasound). Assume that it is in one typical trial. The mass of the subject plus the pallet is. The pallet moves in after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. (This simplified example illustrates the principle of ball is to cardiograph, but in practice a more sophisticated model of heart function is used.)
A small block of mass is released from rest at the top of a frictionless, curve-shaped wedge of mass , which sits on a frictionless, horizontalsurface as shown in Figure P9.80a. When theblock leaves the wedge, its velocity is measured to be to the right as shown in Figure P9.80b.(a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height of the wedge?
Review: After a rubber ball is dropped from a height of 1.75 m, it bounces off a concrete floor and rebounds to a height of .
(a) Determine the magnitude and direction of the impulse delivered to the ball by the floor.
(b) Estimate the time the ball is in contact with the floor and use this estimate to calculate the average force the floor exerts on the ball.
Question: (a) Three carts of masses, , and move on a frictionless, horizontal track with speeds of to the right, to the right, and to the left as shown in Figure P9.34. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts.
(b) What If? Does your answer in part (a) require that all the carts collide and stick together at the same moment? What if they collide in a different order?
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