# Two people of differing masses on a sheet of ice

• aCab
In summary, the conversation discusses a problem involving two men on ice and their resulting velocities after one man pushes the other with a force of 10 Newtons. The book uses the terms acceleration and velocity interchangeably, but the smaller man will actually have a higher velocity due to his lower mass. This can be explained using the impulse formula or the acceleration formula.
aCab
Okay, so I just recently purchased a book from Barnes and Nobles on Physics. I've taken a university level Physics course, so I have a general knowledge of some concepts.

I was reading something that I'm not sure holds water. Here's the problem:

Two men are standing on a sheet of ice. The larger of the two men is a giant weighing 300 kg while the smaller man reaches out and pushes the the larger man with all of his might--a total force of 10 Newtons. What will be the result of this action? Which of the two will slide across the ice and with what velocity?

The book goes on to calculate the accelerations of both parties using F = ma. The acceleration for the bigger man is 0.333 m/s^2 and 0.1 m/s^2 for the smaller man. But they lose me because they use the words acceleration and velocity interchangeably. They say that the "larger man will slide backward with a velocity of 0.333 m/s while the smaller man will slide at 0.1 m/s." If this is the case, wouldn't the larger man slide faster? The conclusion that the book wants to get at is that the smaller man will move much easier on the ice, but how do you justify this using acceleration?

Welcome to PF, aCab.
For starters, the force on the men will be equal (Newton's 3rd law).
The force will act for the same time on both men.
There is no mention of the friction force, so I will use zero friction - not unreasonable for ice. So only the 10 N force on each man.
The impulse formula F*Δt = m*Δv solved for Δv = F*Δt/m let's us compare the velocities attained. F and Δt are the same for both men so only the m varies, and the equation says Δv is larger for the less massive man.

You can see with with acceleration formulas as well. F = m*a,
a = F/m says the guy with the smaller mass has the larger acceleration and so will gain the higher velocity.

## 1. How does the difference in mass affect the movement of two people on a sheet of ice?

The difference in mass between two people on a sheet of ice can affect their movement in a few different ways. The person with a larger mass will typically have a greater force of gravity acting on them, making it easier for them to push off and move across the ice. However, this also means they will have a harder time stopping or changing direction. On the other hand, the person with a smaller mass will have less force of gravity acting on them, making it more difficult for them to move across the ice but also easier for them to stop or change direction.

## 2. Why is it easier for a person with a larger mass to move on ice compared to a person with a smaller mass?

As mentioned before, a person with a larger mass will have a greater force of gravity acting on them. This means that when they push off the ice, they will have more force behind their movement, making it easier for them to glide across the ice. Additionally, a person with a larger mass will have more friction between their body and the ice, providing more traction and stability as they move.

## 3. What happens when two people of equal mass are on a sheet of ice?

If two people of equal mass are on a sheet of ice, their movement will be fairly balanced. They will have similar forces of gravity acting on them, making it easier for them to push off and move across the ice. However, they may still experience some difficulty stopping or changing direction due to the friction between their bodies and the ice.

## 4. How does the surface area of a person's body affect their movement on ice?

The surface area of a person's body can also affect their movement on ice. A person with a larger surface area, such as a wider stance or longer arms, will have more friction between their body and the ice. This can make it easier for them to stop or change direction, but also more difficult for them to move across the ice. Conversely, a person with a smaller surface area will have less friction and may find it easier to move across the ice, but may also have a harder time stopping or changing direction.

## 5. What other factors besides mass and surface area can affect a person's movement on ice?

Other factors that can affect a person's movement on ice include the temperature of the ice, the texture of the ice, and the type of footwear they are wearing. Warmer ice will have less friction, making it easier to move across, while colder ice will have more friction and may be more difficult to move on. Additionally, rough or uneven ice can also impact a person's movement. The type of footwear worn, such as ice skates or regular shoes, can also affect a person's traction and stability on the ice.

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