Speed and magnitude of velocity

[mit_hacker]

Homework Statement

Hello everyone! There is this one question that I do not have the answer to so I was just wondering if anyone could check my solutions and answers and let me know whether I was correct or not. Thank-you for your time!

Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his cart, m_cart, is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest.

Chuck then picks up a ball of mass m_ball and throws it to Jackie, who catches it. Assume that the ball travels in a straight line parallel to the ground (ignore the effect of gravity). After Chuck throws the ball, his speed relative to the ground is v_c. The speed of the thrown ball relative to the ground is v_b.

Jackie catches the ball when it reaches her, and she and her cart begin to move. Jackie's speed relative to the ground after she catches the ball is v_j.

When answering the questions in this problem, keep the following in mind:

1. The original mass m_cart of Chuck and his cart does not include the mass of the ball.
2. The speed of an object is the magnitude of its velocity. An object's speed will always be a nonnegative quantity.

Homework Equations

The Attempt at a Solution

(a) The relative VELOCITY will be the difference in the two velocities (i.e.) vb-va. But since we are asked the speed of them, we will have to add them up because they are in oppositte direction so actually, the velocity will also be vb+va.

(b) By law of conservation of momentum, (mcart)(vcart) = (mball)(vball). But I don't know how to bring u into the picture.

(c) Same problem here!

(d) By law of conservation of momentum,

vj = (mcart)(vcart) / (mj)

(e) Same problem here!

Please help me. Thanks a lot for your time and effort!

 

[Astronuc]

How about a relationship between kinetic energies

(mcart)(vcart)² and (mball)(vball)².

 

[m2003]

Hello! It looks like you have a good understanding of the concepts involved in this problem. To address your specific questions:

(a) The relative velocity of Chuck and Jackie will indeed be the difference between their velocities, as you mentioned. However, since we are dealing with speeds (magnitudes of velocity), we need to take the absolute value of this difference. So the relative speed will be |vb - va|.

(b) The law of conservation of momentum tells us that the total momentum before and after an interaction must be equal. In this problem, the momentum before Chuck throws the ball is 0, since both he and Jackie are at rest. After Chuck throws the ball, the total momentum is (m_ball)(v_ball), since the ball is the only object in motion. This momentum must be conserved, so after Jackie catches the ball, the total momentum will still be (m_ball)(v_ball). We can then use this to find the speed of Jackie and her cart after the interaction.

(c) Similarly to part (b), we can use the law of conservation of momentum to find the speed of Chuck and his cart after the interaction. The total momentum before the ball is thrown is 0, and after the ball is thrown it is (m_ball)(v_ball). We can then use this to find the speed of Chuck and his cart.

(d) To find the speed of Jackie after catching the ball, we can use the law of conservation of momentum again. This time, the total momentum before the interaction is (m_cart)(v_cart), and after the interaction it is (m_cart + m_ball)(v_j). We can set these two equal and solve for v_j.

(e) To find the speed of Chuck after throwing the ball, we can use the same approach as in part (d). The total momentum before the interaction is (m_cart)(v_cart), and after the interaction it is (m_cart + m_ball)(v_c). We can set these two equal and solve for v_c.

I hope this helps and clarifies your understanding of the problem. Keep up the good work!