Problem 28 - College Physics

Question

amateur skater of mass \(M\) (when fully dressed) is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by removing her gloves of mass \(m\) and throwing them in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a velocity \(\vec{v}_{g b r v a}\). Explain whether or not she moxes. If she does move, calculate her velocity \(\vec{v}_{\text {gill }}\) relative to the Farth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her.

Step-by-Step Solution

Answer

Yes, the skater does move. Her velocity relative to the Earth after she throws the gloves is \( \vec{v}_{\text {gill }} = - \frac{m \vec{v}_{g b r v a}} {M} \). From the force perspective, she moves because the force exerted on her by the gloves (equal and opposite to the force she exerts on the gloves) sets her into motion.

Step 1: Understand the concept of momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. It's important to know that momentum is conserved in an isolated system, meaning that the total momentum before an event must be equal to the total momentum after the event. In this case, let's consider the skater and her gloves as the isolated system.

Step 2: Apply the conservation of Momentum in the beginning

At the beginning, neither the skater nor her gloves are moving, so the total momentum is zero.

Step 3: Apply conservation of Momentum after throwing the gloves

After throwing the gloves, the total momentum must still be zero. The momentum of the gloves is \( m \vec{v}_{g b r v a} \) (mass times velocity), so the momentum of the skater must be \( - m \vec{v}_{g b r v a} \) (the minus sign indicates that the momentum is in the opposite direction, as the skater moves in the opposite direction of the gloves). Since the momentum of the skater is also \( M \vec{v}_{\text {gill }} \), we can equate the two and solve for \( \vec{v}_{\text {gill }} \).

Step 4: Calculate her velocity

Set \( M \vec{v}_{\text {gill }} = - m \vec{v}_{g b r v a} \), solve for \( \vec{v}_{\text {gill }} \), we get \( \vec{v}_{\text {gill }} = - \frac{m \vec{v}_{g b r v a}} {M} \). The negative sign indicates that the skater moves in the opposite direction of the gloves.

Step 5: Discuss her motion

From the force perspective, when the skater throws her gloves, she exerts a force on the gloves, and the gloves exert an equal and opposite force on her (according to Newton's third law), causing her to move in the opposite direction. Because of the negligible friction on the ice rink, this force will set her into motion.

Key Concepts

Newton's Third Law

Newton's third law states that for every action, there is an equal and opposite reaction. This principle is crucial in understanding how the skater moves across the ice. When the skater throws the gloves, she applies a force to them. According to Newton's third law, the gloves apply an equal force back onto her. This reaction force propels the skater in the opposite direction to the gloves.

Imagine standing on a skateboard and throwing a ball; as you throw the ball, you feel a backward push—this is the same concept!

• Action: Skater throws gloves forward.
• Reaction: Gloves push skater backward.

The ice being nearly frictionless allows this reaction force to move the skater significantly, since no other forces are acting to slow her down.

Momentum

Momentum is a measure of an object's motion and is the product of its mass and velocity. The equation is given by \( p = m imes v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. In any isolated system, momentum is conserved. This means that the total momentum before any event (like the skater throwing the gloves) is equal to the total momentum after the event.

In this scenario, neither the skater nor her gloves initially have momentum because they are at rest. However, after the throw, the skater has a momentum in one direction and the gloves in another, equal but opposite direction. The equation becomes: \( M \vec{v}_{\text{gill}} = - m \vec{v}_{gbrva} \), leading to the skater's velocity \( \vec{v}_{\text{gill}} = -\frac{m \vec{v}_{gbrva}}{M} \).
The negative sign indicates the opposite direction to the gloves. Total momentum remains zero, fulfilling the conservation of momentum.

Frictionless Surface

A frictionless surface allows the skater to move freely once she throws the gloves. Ice rinks are often used to approximate such surfaces due to their low friction. Without significant frictional forces, even small forces can result in noticeable motion.

On a rough surface, the force from throwing gloves might be absorbed by friction, leaving the skater mostly stationary. However, the ice reduces these hindering forces significantly. This allows the reaction force from throwing the gloves to translate efficiently into motion.

• Ice supports smooth motion with little resistance.
• Allows small forces to cause larger movements.

This condition is crucial for illustrating the conservation of momentum and Newton's third law in a clear way, as the absence of friction simplifies the problem considerably.