Cart A, of mass 3.00 kg, approaches and collides with cart B, which has a mass of 7.00 kg and is initially at rest.
When the springs have reached their maximum compression,
a. cart A has come to rest relative to the ground.
b. both carts have the same velocity.
c. both carts have the same momentum.
d. all the initial kinetic energy of cart A has been converted to elastic potential energy.
b. both carts have the same velocity. Just did on MP
To answer this question, let's analyze the conservation of momentum and kinetic energy during the collision of the two carts.
1. Conservation of momentum:
According to the law of conservation of momentum, the total momentum of an isolated system is conserved before and after a collision. In this case, the system consists of cart A and cart B.
Before the collision, cart A has a nonzero velocity, while cart B is initially at rest. Therefore, the initial momentum of the system is solely due to cart A's momentum.
p(initial) = m_A * v_A
After the collision, both carts are in contact and move together as a single system. Let's assume that they have a final velocity v_f.
p(final) = (m_A + m_B) * v_f
Since the total momentum is conserved, we can set the initial and final momentum equal to each other:
p(initial) = p(final)
m_A * v_A = (m_A + m_B) * v_f
Based on this equation, we can already determine that both carts will have the same final velocity (v_f) because the masses of the carts are known.
2. Conservation of kinetic energy:
The law of conservation of kinetic energy states that the total kinetic energy of an isolated system remains constant before and after a collision.
Before the collision, only cart A has kinetic energy,
KE(initial) = (1/2) * m_A * v_A^2
After the collision, the kinetic energy is shared between both carts. Additionally, some of the kinetic energy is converted to elastic potential energy stored in the compressed springs between the carts.
KE(final) = (1/2) * (m_A + m_B) * v_f^2 + U_elastic
Since the total kinetic energy is conserved, we can equate the initial and final kinetic energy:
KE(initial) = KE(final)
(1/2) * m_A * v_A^2 = (1/2) * (m_A + m_B) * v_f^2 + U_elastic
From this equation, we can see that some of the initial kinetic energy is converted into elastic potential energy stored in the compressed springs (U_elastic). Therefore, option d, where all the initial kinetic energy of cart A has been converted to elastic potential energy, is correct.
In summary, the correct answer is:
d. All the initial kinetic energy of cart A has been converted to elastic potential energy.