Question

A 48.8 kg girl is standing on a 136 kg plank. The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless surface. The girl begins to walk along the plank at a constant velocity of 1.59 m/s relative to the plank.

What is her velocity relative to the ice surface?

What is the velocity of the plank relative to the ice surface?

Answer 1

Momentum is conserved.

Initial momentum = 0 (because she's at rest).
Final momentum = 0 (because it has to equal initial momentum)

Since the final momentum (while she's walking) is zero, this means the girl's momentum in one direction must exactly cancel out the plank's momentum in the opposite direction.

Let v_g = girl's velocity relative to ice.
Let v_p = plank's velocity relative to ice.

total momentum = 0
(48.8kg)(v_g) + (136kg)(v_p) = 0

We also know that the girl's velocity relative to the PLANK is 1.59 m/s. You get relative velocity by subtracting one velocity from another. So:

Girl's velocity relative to plank = 1.59 m/s
v_g ? v_p = 1.59 m/s

To recap, you have these two equations:

(48.8kg)(v_g) + (136kg)(v_p) = 0
v_g ? v_p = 1.59 m/s

That's 2 equations in 2 unknowns; so you can use algebra to solve for v_g and v_p.

v_g= 1.17 m/s

v_p= - 0.4198 m/s

Answer 2

Momentum is conserved.

Initial momentum = 0 (because she's at rest).
Final momentum = 0 (because it has to equal initial momentum)

Since the final momentum (while she's walking) is zero, this means the girl's momentum in one direction must exactly cancel out the plank's momentum in the opposite direction.

Let v_g = girl's velocity relative to ice.
Let v_p = plank's velocity relative to ice.

total momentum = 0
(48.8kg)(v_g) + (136kg)(v_p) = 0

We also know that the girl's velocity relative to the PLANK is 1.59 m/s. You get relative velocity by subtracting one velocity from another. So:

Girl's velocity relative to plank = 1.59 m/s
v_g ? v_p = 1.59 m/s

To recap, you have these two equations:

(48.8kg)(v_g) + (136kg)(v_p) = 0
v_g ? v_p = 1.59 m/s

That's 2 equations in 2 unknowns;use algebra to solve for v_g and v_p. (v_g is the answer to Part 1; v_p is the answer to Part 2.

v_g = 1.17

v_p = -0.419