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A 45.8-kg girl is standing on a 151-kg plank. Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity of 1.46 m/s relative to the plank.

(a) What is her velocity relative to the surface of the ice?
(b) What is the velocity of the plank relative to the surface of the ice?

Answer :

Answer:

a. 1.12m/s

b. 0.34m/s

Explanation:

Let Vgi = the velocity of the girl relative to the ice

Let Vgp = the velocity of the girl relative to the plank

Let Vpi = the velocity of the plank relative to the ice

Vgi = Vgp + Vpi

From the question, Vgp = 1.46/s

So, Vgi = 1.46 + Vpi

Conservation of Momentum (Relative to the ice), we have

Initial Momentum = Final Momentum

Because the girl and the plank are at rest initially, the initial Momentum = 0

So, 0 = Mg * Vgi + Mp * Vpi

Where Mg = Mass of the girl = 45.8kg

Mo = Mass of the Plank = 151kg

Make Vpi the subject of the formula, we then have

-Mg*Vgi = Mp*Vpi ---------- Divide through by Mp

Vpi = -Mg * Vgi/Mp

Vpi =( -45.8 * Vgi)/151

Vpi = -45.8Vgi/151

Remember that, we have (Vgi = 1.46 + Vpi)

We then substitute the expression of Vpi in the above equation

That is;

Vgi = 1.46 + (-45.8Vgi/151) ------- Open the bracket

Vgi = 1.46 - 45.8Vgi/151 ----------- Multiply through by 151

151 * Vgi = 151 * 1.46 - 45.8Vgj

151Vgi = 220.46 - 45.8Vgi --------- Collect like terms

151Vgi + 45.8Vgi = 220.46

196.8Vgi = 220.46 --------------- Divide through by 196.8

Vgi = 220.46/196.6

Vgi = 1.1213631739572736

Vgi = 1.12 m/s (Approximated)

So, the velocity of the girl relative to the ice is 1.12m/s

b. Velocity of the plank, relative to the ice

We can solve this using (Vgi = 1.46 + Vpi)

All we need to do is substitute 1.12 for Vgi in the equation

So, we have

Vgi = 1.46 + Vpi becomes

1.12 = 1.46 + Vpi -------- Collect like terms

1.12 - 1.46 = Vpi

-0.34 = Vpi

So, the Velocity of the plank is 0.34m/s to the left

Answer: (a) 2.095 m/s (b) 0.635 m/s

Explanation: The initial momentum is = 0. Also, the final momentum is = 0. At the initial momentum she is at rest hence it is = 0. Whereas, the final momentum is = 0 because it has to equal the initial momentum.

This helps us to understand that when final momentum is = 0 then the girl's direction would cancel the affect of the plank's momentum in the opposite direction.

Therefore,

Vg = Velocity of girl

Vp= Velocity of plank

(45.8)(Vg) + (151)(Vp) = 0

Girl's velocity relative to the plank is 1.46 m/s

Therefore, Vg - Vp = 1.46

Vg = 1.46 + Vp

(45.8)(1.46 + Vp) = 151Vp

66.868 + 45.8Vp = 151Vp

66.868 = 105.2Vp

Vp = 0.635 m/s

Vg = 1.46 + 0.635

Vg = 2.095

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