Question

Chuck and Jackie stand on separate carts, both of which can slide without friction. The combined mass of Chuck and his car, mcartmcart, is identical to the combined mass of Jackie and her cart. Initially, Chuck and Jackie and their carts are at rest.
Chuch then picks up a ball of mass mballmball 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 vc. The speed of the thrown ball relative to the ground is vb. 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 vj.

When answering the questions in this problem, keep the following in mind:
1. The original mass mcartmcart 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 non-negative quantity.

Required:
a. Find the relative speed u between Chuck and the ball after Chuck has thrown the ball.
b. What is the speed vb of the ball (relative to the ground) while it is in the air?
c. What is Chuck's speed vc (relative to the ground) after he throws the ball?
d. Find Jackie's speed vj (relative to the ground) after she catches the ball in terms of vb.
e. Find Jackie's speed vj (relative to the ground) after she catches the ball in terms of u.

Answer 1

Explanation:

a.)

We find the relative speed

u = vb + vc

b.)

chuck and the cart are at a rest position

mcartvc = mballvb

from part a above,

vc = u - vb

mcartu = vb(mcart + mball)

make vb the  subject of the equation

$vb=\frac{mcartu}{mcart +mball}$

c.)

from anser a,

vb = u - vc

then mcart vc = mball(u-vc)

$vc= \frac{Mballu}{Mcart+Mball}$

d.

Mballvb = (Mcart + Mball)vj

we make vj the subject to get her relative speed

$vj=\frac{MballVb}{Mcart+Mball}$

e.

given the solution in part d above,

we have

$vj=\frac{MballVb}{Mcart+Mball}$

remember,

vb = u - vc

such that

$Vj = \frac{Mball(u-v)}{Mball +Mcart}$

thank you!