A 45.0 kg girl is standing on a 163 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 1.45 m/s relative to the plank.
(a) What is her velocity relative to the surface of ice?
(b) What is the velocity of the plank relative to the surface of ice?
If the girl is walking at 1.45 m / s in the x direction relative to the plank and the plank is moving at velocity V in the x direction then her total velocity is 1.45 + V in the x direction. (note - V better come out negative, in the negative x direction in part b)
There are no external horizontal forces on this system.
Therefore there will be no change in the initial horizontal momentum, which is zero.
final momentum:
0 = 45 * (1.45+V) + 163 V
0 = 65.25 + 45 V + 163 V
208 V = - 65.25
V = -0.314 m/s
by the way that means the girl is going at 1.45 -.314 = 1.14 relative to the ice and as a check in the ice coordinate system
0 = -0.314 * 163 + 1.14 * 45 ???
= -51.2 + 51.3 close enough
(a) Her velocity relative to the surface of ice is the same as her velocity relative to the plank, which is 1.45 m/s. So she's sliding along the ice at a cool speed!
(b) The velocity of the plank relative to the surface of ice remains zero. Since there's no external force acting on the system, the total momentum of the girl-plank system is conserved. Since the girl and plank move together, their combined mass remains at rest and there's no change in velocity for the plank. So the plank is just chilling on the ice, not going anywhere.
To find the answers, we can use the concept of conservation of momentum. According to the law of conservation of momentum, the total momentum before an event is equal to the total momentum after the event.
Let's assume the positive direction is to the right.
(a) To find the girl's velocity relative to the surface of the ice, we can use the equation:
Total initial momentum = Total final momentum
The initial momentum is zero since both the girl and the plank are at rest. The final momentum is the sum of the momenta of the girl and the plank.
Final momentum = (Mass of the girl × Velocity of the girl) + (Mass of the plank × Velocity of the plank)
Final Momentum = (45.0 kg × 1.45 m/s) + (163 kg × Velocity of the plank)
Since the plank and the girl move at the same constant velocity relative to each other, the velocity of the plank relative to the surface of the ice is also 1.45 m/s.
Final Momentum = (45.0 kg × 1.45 m/s) + (163 kg × 1.45 m/s)
Now we can solve for the velocity of the plank:
Final Momentum = (45.0 kg + 163 kg) × Velocity of the plank
Final Momentum = 208.0 kg × Velocity of the plank
Now we can solve for the velocity of the plank:
Velocity of the plank = Final Momentum / 208.0 kg
(b) Now that we know the velocity of the plank is 1.45 m/s, we can conclude that the velocity of the girl relative to the surface of the ice is also 1.45 m/s.
To find the velocity of the girl relative to the surface of ice, we need to consider her velocity relative to the plank and the velocity of the plank relative to the surface of ice.
(a) Velocity of the girl relative to the surface of ice:
The velocity of the girl relative to the plank is given as 1.45 m/s. Since the plank is initially at rest (velocity relative to ice = 0), the girl's velocity relative to the surface of ice would be the same as her velocity relative to the plank: 1.45 m/s.
(b) Velocity of the plank relative to the surface of ice:
To find the velocity of the plank relative to the surface of ice, we can use the concept of conservation of momentum. The total momentum before the girl starts walking must be the same as the total momentum after she starts walking, since there are no external forces on the system.
Before the girl starts walking, the total momentum is zero since both the girl and the plank are at rest.
After the girl starts walking, she exerts a backward force on the plank, pushing it forward. According to Newton's third law, the plank exerts an equal and opposite force on the girl, causing her to move forward. This pair of forces does not change the total momentum of the system.
Therefore, the velocity of the plank relative to the surface of ice would also be zero.