(a)
The final velocity of the person and the cart.
(a)
Answer to Problem 49P
The final velocity of the person and the cart is
Explanation of Solution
Write the expression for conservation of momentum for inelastic collision.
Here,
Conclusion:
Substitute
In vector notation the velocity of the cart and person is,
Therefore, the final velocity of the person and the cart is
(b)
The friction force acting on the person.
(b)
Answer to Problem 49P
The friction force acting on the person is
Explanation of Solution
Write the expression for normal force by using Newton’s second law in y direction
Here,
Write the expression for frictional force exerted on the person.
Here,
Conclusion:
Substitute
Substitute
In vector notation the frictional force is,
Therefore, the friction force acting on the person is
(c)
The time taken for the frictional force acting on the person.
(c)
Answer to Problem 49P
The time taken for the frictional force acting on the person is
Explanation of Solution
Write the expression for person’s momentum equal to the impulse.
Here,
Write the expression for initial momentum.
Here,
Write the expression for final momentum.
Here,
Write the expression for impulse.
Here,
Conclusion:
Substitute the equations (V), (VI) and (VII) in equation (IV).
Substitute
Therefore, the time taken for the frictional force acting on the person is
(d)
The change in momentum of the person and the cart.
(d)
Answer to Problem 49P
The change in momentum of the person is
Explanation of Solution
Write the expression for change in momentum of the person.
Write the expression for change in momentum of the cart.
Conclusion:
Substitute
Substitute
Therefore, the change in momentum of the person is
(e)
The displacement of the person relative to the ground.
(e)
Answer to Problem 49P
The displacement of the person relative to the ground is
Explanation of Solution
Write the expression for displacement of the person relative to the ground.
Here,
Conclusion:
Substitute
Therefore, the displacement of the person relative to the ground is
(f)
The displacement of the cart relative to the ground.
(f)
Answer to Problem 49P
The displacement of the cart relative to the ground is
Explanation of Solution
Write the expression for displacement of the cart relative to the ground.
Conclusion:
Substitute
Therefore, the displacement of the cart relative to the ground is
(g)
The change in kinetic energy of the person.
(g)
Answer to Problem 49P
The change in kinetic energy of the person is
Explanation of Solution
Write the expression for change in kinetic energy of the person.
Conclusion:
Substitute
Therefore, the change in kinetic energy of the person is
(h)
The change in kinetic energy of the cart.
(h)
Answer to Problem 49P
The change in kinetic energy of the cart is
Explanation of Solution
Write the expression for change in kinetic energy of the cart.
Conclusion:
Substitute
Therefore, the change in kinetic energy of the cart is
(i)
Why the answers part (g) and (h) are differ.
(i)
Answer to Problem 49P
Because. the distance moved by the cart is different from the distance moved by the point of application of friction force to the cart.
Explanation of Solution
The force exerted by the person on the cart must be equal in magnitude and opposite in direction to the force exerted by the cart on the person. The changes in momentum of the two objects must be equal in magnitude and must be added to zero.
The change in kinetic energy is different in magnitude and does not add to zero.
Conclusion:
The following situation is represents in two ways,
The distance moved by the cart is different from the distance moved by the point of application of friction force to the cart.
The total change in mechanical energy for both objects add together becomes zero, it is perfectly in elastic collision.
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Chapter 8 Solutions
Principles of Physics: A Calculus-Based Text
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