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Q58P

Expert-verifiedFound in: Page 291

Book edition
9th Edition

Author(s)
Raymond A. Serway, John W. Jewett

Pages
1624 pages

ISBN
9781133947271

A ${\mathbf{60}}{\mathbf{.}}{\mathbf{0}}{\mathbf{-}}{\mathit{k}}{\mathit{g}}$ 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 ${\mathbf{15}}{\mathbf{.}}{\mathbf{0}}{\mathbf{}}{\mathit{c}}{\mathit{m}}$. 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 $102.9\text{kg}\xb7\text{m/s}$.

(d) Yes, it can be said that the momentum came from the floor.

(e) The kinetic energy when the person leaves the floor is $88.24\text{J}$.

(f) The kinetic energy required for the person leave the floor does not come from the floor.

The mass of the person is, $m=60.0\text{kg}$.

The maximum height of the center of mass of the person is, $h=15.0\text{cm}=0.15\text{m}$.

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,

$\Delta KE+\Delta PE=0\phantom{\rule{0ex}{0ex}}K{E}_{i}+P{E}_{i}=K{E}_{f}+P{E}_{f}$

At maximum height, the final velocity of the person is 0 and at the initial position the person is on the ground. So, putting $K{E}_{f}=0$ and $P{E}_{i}=0$ in the equation,

$\frac{1}{2}m{{v}_{i}}^{2}+0=0+mgh\phantom{\rule{0ex}{0ex}}{v}_{i}=\sqrt{2gh}\phantom{\rule{0ex}{0ex}}{v}_{i}=\sqrt{2\times 9.8{\text{m/s}}^{2}\times 0.15\text{m}}\phantom{\rule{0ex}{0ex}}{v}_{i}=1.715\text{m/s}$

The value of the momentum when the person leaves the floor is given by,

$P=m{v}_{i}\phantom{\rule{0ex}{0ex}}P=60\text{kg}\times 1.715\text{m/s}\phantom{\rule{0ex}{0ex}}P=102.9\text{kg}\xb7\text{m/s}$

Hence, the momentum when the person leaves the floor is $102.9\text{kg}\xb7\text{m/s}$.

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,

$KE=\frac{1}{2}m{{v}_{i}}^{2}\phantom{\rule{0ex}{0ex}}KE=\frac{1}{2}\times 60\text{kg}\times {\left(1.715\text{m/s}\right)}^{2}\times \frac{1\text{J}}{1\text{kg}\xb7{\text{m}}^{2}{\text{/s}}^{2}}\phantom{\rule{0ex}{0ex}}KE=88.24\text{J}$

Hence, the kinetic energy when the person leaves the floor is $88.24\text{J}$.

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.

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