A 43.3 kg girl is standing on a 125 kg plank.

The plank, originally at rest, is free to slide on
a frozen lake, which is a flat, frictionless sup-
porting surface. The girl begins to walk along
the plank at a constant speed of 1.87 m/s to
the right relative to the plank.
What is her velocity relative to the ice sur-
face?
Answer in units of m/s

What is the velocity of the plank relative to
the ice surface?
Answer in units of m/s

To determine the velocity of the girl relative to the ice surface, we need to take into account the velocity of the plank on which she is standing. Since she is walking on the plank, her velocity relative to the plank is given as 1.87 m/s to the right.

To find her velocity relative to the ice surface, we need to calculate the net velocity by adding the velocity of the plank.

However, before doing so, let's determine the direction of the velocities. Let's assume moving to the right is considered positive and moving to the left is negative.

Given:
Girl's mass, m_g = 43.3 kg
Plank's mass, m_p = 125 kg
Girl's velocity relative to the plank, v_gp = 1.87 m/s to the right

The total momentum of the system is conserved, which means the momentum before the girl starts walking must be equal to the momentum after she starts walking.

The initial momentum before the girl starts walking is zero since both the girl and the plank are at rest.

The final momentum after the girl starts walking can be calculated by considering the mass and velocity of the girl and the plank.

Girl's momentum relative to the ice surface = m_g * v_gs (where v_gs is the velocity of the girl relative to the ice surface)
Plank's momentum relative to the ice surface = m_p * v_ps (where v_ps is the velocity of the plank relative to the ice surface)

Since momentum is conserved, the initial and final momentum must be equal. Therefore, we have:

Initial Momentum = Final Momentum

0 = m_g * v_g + m_p * v_p

Solving for v_gs (velocity of the girl relative to the ice surface):

v_gs = -(m_p / m_g) * v_ps

Now we have the equation for the velocity of the girl relative to the ice surface in terms of the velocity of the plank relative to the ice surface (v_ps).

To determine the velocity of the plank relative to the ice surface (v_ps), we can use the same equation, where we substitute v_gs = -1.87 m/s (given velocity of the girl relative to the plank) and solve for v_ps.

Let's substitute the values and calculate the answers:

v_gs = -(m_p / m_g) * v_ps

v_gs = -(125 kg / 43.3 kg) * v_ps

v_gs = -2.885 * v_ps

Since the girl's velocity relative to the plank is given as 1.87 m/s to the right, we substitute that into the equation:

1.87 m/s = -2.885 * v_ps

Solving for v_ps (velocity of the plank relative to the ice surface):

v_ps = 1.87 m/s / (-2.885)

v_ps ≈ -0.648 m/s

This means the velocity of the plank relative to the ice surface is approximately -0.648 m/s to the left.

To find the velocity of the girl relative to the ice surface, we substitute the value of v_ps back into the equation:

v_gs = -(125 kg / 43.3 kg) * (-0.648 m/s)

v_gs ≈ 1.871 m/s

Therefore, the girl's velocity relative to the ice surface is approximately 1.871 m/s to the right. And the velocity of the plank relative to the ice surface is approximately -0.648 m/s to the left.