Review. A 60.0-kg person running at an initial speed of 4.00 m/s jumps onto a 120-kg cart initially at rest (Fig. P9.37). The person slides on the cart’s top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is 0.400. Friction between the cart and ground can be ignored. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the friction force acting on the person while he is sliding across the top surface of the cart. (c) How long does the friction force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. (c) Determine the displacement of the person relative to the ground while he is sliding on the cart. (f) Determine the displacement of the cart relative to the ground while the person is sliding. (g) Find the change in kinetic energy of the person. (h) Find the change in kinetic energy of the cart. (i) Explain why the answers to (g) and (h) differ. (What kind of collision is this one, and what accounts for the loss of mechanical energy)
Figure P9.37
(a)
The final velocity of the person and cart relative to the ground.
Answer to Problem 37AP
The final velocity of the person and cart relative to the ground is
Explanation of Solution
The mass of the person is
Write the expression of conservation of momentum.
Here,
Substitute
Conclusion:
Therefore, the final velocity of the person and cart relative to the ground is
(b)
The frictional force acting on the person while he is sliding.
Answer to Problem 37AP
The frictional force acting on the person while he is sliding is
Explanation of Solution
Write the expression to calculate the frictional force.
Here,
Substitute
The negative sign indicates that the frictional force is acting toward negative x axis.
Conclusion:
Therefore, the frictional force acting on the person while he is sliding is
(c)
The time duration in which the frictional force is acting on the person.
Answer to Problem 37AP
The time duration in which the frictional force is acting on the person is
Explanation of Solution
Write the expression of Impulse-momentum equation.
Here,
Substitute
Conclusion:
Therefore, the time duration in which the frictional force is acting on the person is
(d)
The change in momentum of the person and cart.
Answer to Problem 37AP
The change in momentum of the person and cart is
Explanation of Solution
Write the expression to calculate the change in momentum of the person.
Substitute
Write the expression to calculate the change in momentum of the cart.
Substitute
Conclusion:
Therefore, the change in momentum of the person and cart is
(e)
The displacement of the person relative to the ground during sliding on the cart.
Answer to Problem 37AP
The displacement of the person relative to the ground during sliding on the cart is
Explanation of Solution
Write the expression to calculate the displacement of the person.
Substitute
Conclusion:
Therefore, the displacement of the person relative to the ground during sliding on the cart is
(f)
The displacement of the cart relative to the ground during the person sliding on the cart.
Answer to Problem 37AP
The displacement of the cart relative to the ground during the person sliding on the cart is
Explanation of Solution
Write the expression to calculate the displacement of the cart.
Substitute
Conclusion:
Therefore, the displacement of the cart relative to the ground during the person sliding on the cart is
(g)
The change in kinetic energy of the person.
Answer to Problem 37AP
The change in kinetic energy of the person is
Explanation of Solution
Write the expression to calculate the change in kinetic energy of the person.
Substitute
`
Conclusion:
Therefore, the change in kinetic energy of the person is
(h)
The change in kinetic energy of the cart.
Answer to Problem 37AP
The change in kinetic energy of the cart is
Explanation of Solution
Write the expression to calculate the change in kinetic energy of the cart.
Substitute
`
Conclusion:
Therefore, the change in kinetic energy of the cart is
(i)
The reason due to which the answer in part (g) and (h) are different.
Answer to Problem 37AP
The collision between the person and the cart is perfectly inelastic collision and the loss of energy is due to frictional force.
Explanation of Solution
The force acting on the person must be equal in magnitude and opposite in direction to the force exerted by the cart on the person.
According to the conservation of linear momentum, the change in momentum of the person and the cart must be equal in magnitude and must add to zero. The change in kinetic energy of the person and cart must be equal for elastic collision but in this case change in kinetic energy of the person and cart is not equal, which shows that this is inelastic collision.
The reason of change in kinetic energy of both object of not being same is, the displacement of person and the cart is not same due to frictional force acting on the person.
Due to the frictional force the loss of energy in form of heat is the internal energy.
Write the expression to calculate the internal energy.
Substitute
Thus, the collision between the person and the cart is perfectly inelastic collision due to the loss of energy.
Conclusion:
Therefore, the collision between the person and the cart is perfectly inelastic collision and the loss of energy is due to frictional force.
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Chapter 9 Solutions
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