A girl of mass (Mg) is standing on a plank of mass (Mp). 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 (vgp) to the right relative to the plank. (The subscript gp denotes the girl relative to plank. Use any variable or symbol stated above as necessary.)
(a) What is the velocity vpi of the plank relative to the surface of the ice?
(b) What is the girl's velocity vgi relative to the ice surface?
The total momentum remains zero and the center of mass remains in the original location.
(a) The plank's velocity (relative to ice) is vpi. The girl's velocity relative to ice is vgp + vpi
Mp*vpi + Mg(vgp + vpi) = 0
vpi (Mg + Mp) = -Mg*vgp
vpi = -[Mg/(Mg+Mp)]*vgp
(b) vgi = vgp + vpi
= vgp{1 - [Mg/(Mg+Mp)]}
= vgp*[Mp/(Mg +Mp)]
THANK U
To answer these questions, we can use the principle of conservation of momentum. The total momentum of the system remains constant when there are no external forces acting on it.
Let's start with part (a) and find the velocity vpi of the plank relative to the surface of the ice.
We know that the girl is on the plank, and she is walking along it at a constant velocity (vgp) to the right relative to the plank. The relative velocity between the girl and the plank is vg-gp (girl's velocity minus plank's velocity).
Since there is no external force or friction acting on the plank and the girl, the total momentum of the system remains constant. Initially, both the plank and the girl are at rest, so the total momentum is zero.
To find the final momentum, we need to consider the masses and velocities of the plank and the girl. The plank has a mass (Mp) and its velocity relative to the surface of the ice is vpi. The girl has a mass (Mg) and her velocity relative to the ice surface is vgi.
According to the principle of conservation of momentum, we have:
0 = Mp * vpi + Mg * vgi
Since the girl is walking at a constant velocity (vgp) relative to the plank, we can express her velocity relative to the ice surface (vgi) in terms of the plank's velocity relative to the ice surface (vpi) and the girl's velocity relative to the plank (vgp):
vgi = vgp + vpi
Plugging this into the conservation of momentum equation, we get:
0 = Mp * vpi + Mg * (vgp + vpi)
Now we can solve for vpi:
Mp * vpi = -Mg * (vgp + vpi)
Mp * vpi + Mg * vpi = -Mg * vgp
vpi * (Mp + Mg) = -Mg * vgp
vpi = -Mg * vgp / (Mp + Mg)
So, the velocity vpi of the plank relative to the surface of the ice is given by:
vpi = -Mg * vgp / (Mp + Mg)
Moving on to part (b), we need to find the girl's velocity vgi relative to the ice surface.
We already know that:
vgi = vgp + vpi
Substituting the value of vpi we just found, we get:
vgi = vgp - Mg * vgp / (Mp + Mg)
vgi = vgp * ((Mp + Mg) - Mg) / (Mp + Mg)
vgi = vgp * Mp / (Mp + Mg)
So, the girl's velocity vgi relative to the ice surface is given by:
vgi = vgp * Mp / (Mp + Mg)
To summarize:
(a) The velocity vpi of the plank relative to the surface of the ice is given by:
vpi = -Mg * vgp / (Mp + Mg)
(b) The girl's velocity vgi relative to the ice surface is given by:
vgi = vgp * Mp / (Mp + Mg)