A 60.0-kg person bends his knees and then jumps straight up....

A $60.0-\mathrm{kg}$ 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 15.0 cm . 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.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, which increases as the object is raised to a higher altitude. For a jumper, the maximum height reached corresponds to the point where all the kinetic energy provided at takeoff has been converted into gravitational potential energy. This relationship is central to energy conservation analyses in projectile motion.

Energy Transfer and Source Identification

Energy transfer in the context of a jump involves discerning the origin of the kinetic energy that propels the person upward. Although the floor imparts an impulse through the normal force, it does not contribute energy in the form of work on the person. Instead, the energy comes from the jumper’s internal conversion of chemical energy into mechanical work, highlighting the difference between an external force providing momentum and the actual energy source.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion, calculated as one half of its mass times the square of its velocity. In jumping, the kinetic energy just after takeoff is converted into gravitational potential energy as the jumper ascends. This conversion helps in determining the initial velocity and momentum imparted during the jump.

Impulse

Impulse is defined as the product of a force and the time interval over which it acts, or equivalently, as the integral of the force with respect to time. It is directly related to the change in momentum of an object, meaning that any force that acts over a time period, such as the force exerted by the floor on a person during takeoff, imparts an impulse that changes the person’s momentum.

Momentum

Momentum is a measure of the quantity of motion an object possesses, calculated as the product of its mass and velocity. In the context of jumping, the momentum of the person immediately after leaving the floor is the result of the impulse he receives. Momentum is a vector quantity and is conserved in isolated systems; however, interactions like that between the person and the floor involve external forces and impulses.

Work

Work is the product of a force applied to an object and the displacement in the direction of the force. It represents the transfer of energy to or from an object. The concept is critical when analyzing whether the floor or the person's internal muscles have provided the energy that eventually manifests as the kinetic energy of the jump.

Transcript

00:02 That's standing on the floor and the bend their knees to jump up and they do reach a certain height of 15 centimeters and that person weighs about 60 kilograms um so it has a series of questions asking you to figure out what's going on with this is so the first thing they ask you is does the floor impart any impulse on this person here.

00:32 So you know that if they're standing on this floor here, there is the person exerts a force downwards.

00:41 That is mg.

00:44 But then the floor also exerts a force upwards, and that's the normal force.

00:49 And as this person is jumping here, this force here is acting on this person for a certain amount of time.

00:59 So the reaction to the changes in motion that he's trying to do here will take a certain amount of time, which means that there is an impulse that is equal to a force at a certain time here.

01:17 So as they're jumping, they are exerting a larger force on the ground, which in this case is going to cause a larger reaction on them, which would mean that you have a force term here that's actually a certain amount of time.

01:35 So you do have an impulse on the system.

01:38 So that's the answer to okay.

01:40 So the answer would be yes.

01:44 And then the next part is saying, does the floor do work on the person? so we know that work is force times the distance.

01:59 So if the floor is going to do work on the system, then the force provide a floor has to move a certain distance here.

02:08 And we know that the force is not going to move any distance here.

02:14 So this is from just the definition of work.

02:17 We can see that that's equal to zero because that term is zero.

02:22 So essentially you can determine that no, the first floor is not doing any work on the system.

02:29 Because there's no movement related to the direction of that force...