How Does Momentum Conservation Affect Sled Speed When Susan Jumps On and Off?

[gkangelexa]

Homework Statement

An empty sled is sliding on frictionless ice when Susan drops vertically from a tree above onto the sled. When she lands, does the sled speed up, slow down, or keep the same speed?
The sled slows down.
Later, susan falls sideways off the sled. When she drops off, does the sled speed up, slow down, or keep the same speed?

The Attempt at a Solution

I would think that when she falls off the sled, the speed of the sled should increase because m₂ got smaller, so to conserve momentum, v₂ should get larger
m₁v₁ = m₂v₂


however, the answer is that the sled keeps the same speed... why?

 

[PeterO]

Homework Statement

An empty sled is sliding on frictionless ice when Susan drops vertically from a tree above onto the sled. When she lands, does the sled speed up, slow down, or keep the same speed?
The sled slows down.
Later, susan falls sideways off the sled. When she drops off, does the sled speed up, slow down, or keep the same speed?

The Attempt at a Solution

I would think that when she falls off the sled, the speed of the sled should increase because m₂ got smaller, so to conserve momentum, v₂ should get larger
m₁v₁ = m₂v₂


however, the answer is that the sled keeps the same speed... why?

Remember she fell sideways off the sled, taking her momentum with her.

 

[gkangelexa]

Remember she fell sideways off the sled, taking her momentum with her.

Would she not have "taken her momentum with her" if she had fallen off the back instead of sideways?

 

[PeterO]

Would she not have "taken her momentum with her" if she had fallen off the back instead of sideways?

Yes she would. BUT if she took her life in her hands, stood up, then kicked forward on the sled so that she "left the sled" with zero momentum, the sled would have continued with all the momentum - back to its original speed.

That "kicking off" is the way people commonly dismount from a skate board, on which they are already standing.

Peter

By the way, you were only asked what would happen if she fell out sideways!

 

[rwisz]

It is important to understand the concept of momentum and how it is conserved in a system. In this scenario, we have a sled and a person (Susan) dropping onto the sled. The initial momentum of the system is zero, since the sled is empty and not moving. When Susan drops onto the sled, her momentum is transferred to the sled, causing it to move with a certain speed. This is known as the conservation of momentum.

When Susan falls off the sled, the system is now just the sled. Since there are no external forces acting on the system, the total momentum of the system must remain constant. This means that the momentum gained by the sled when Susan landed on it must be equal to the momentum lost when she falls off. Since the sled has a larger mass compared to Susan, the decrease in velocity of the sled will be smaller compared to the increase in velocity when she fell onto it. Therefore, the sled will keep the same speed.

In terms of the equation m1v1 = m2v2, the mass of the sled (m2) is much larger than the mass of Susan (m1), so even though v1 decreases, the decrease is smaller compared to the increase in v2, resulting in the sled maintaining its speed.

It is important to note that this is an idealized scenario and in reality, there may be other factors at play such as air resistance or friction on the ice. However, the concept of conservation of momentum still holds true in this scenario.