### Video Transcript

A skater of mass 40 kilograms is carrying a box of mass 5.0 kilograms. The skater has a speed of 5.0 metres per second with respect to the floor and is gliding without any friction on a smooth surface. Find the momentum of the box with respect to the floor. Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface.

Let’s start by highlighting some of the important information we’re given. We’re told the mass of the skater is 40 kilograms and that she’s carrying a box of mass 5.0 kilograms. The skater is moving at a speed of 5.0 metres per second. And in part one, we want to solve for the momentum of the box while it’s still being carried by the skater; we will call that 𝑝 sub 𝑐. And in part two, we want to solve for the momentum of the box with respect to the floor after the box has been put down on the perfectly smooth skating surface; we will call that 𝑝 sub 𝑓.

Let’s draw a diagram of the scenario. In the first part of the problem, illustrated in number one, we see the skater holding the box and skating across the ice moving at a speed 𝑣 given as 5.0 metres per second. In the second scenario, part two, the skater has put the box down on the ice, while continuing to move to the right with the same speed 𝑣. In both instances, we want to know what the momentum of the box is. Let’s start by recalling the definition of the momentum of an object.

In its scaler formulation, the momentum 𝑝 of an object is equal to the object’s mass times its speed 𝑣. In the case of the box being carried by the skater, 𝑝 sub 𝑐, the momentum of the box while it’s being carried, is equal to the box’s mass, 𝑚 sub 𝑏, multiplied by the speed of the skater as she carries the box with her, 𝑣. Plugging in 5.0 kilograms for 𝑚 sub 𝑏 and 5.0 metres per second for 𝑣, the momentum of the box while it’s being carried is equal to 25 kilograms metres per second. That’s the box’s momentum as the skater carries it.

Now, what about the momentum of the box when it’s on the floor, 𝑝 sub 𝑓? We’ll use the same basic relationship to solve for its momentum. The momentum of the box on the floor 𝑝 sub 𝑓 is equal to the mass of the box multiplied by its speed. Here, because the surface that the skater slides along is being treated as frictionless, we know that the box will lose no energy and thus no speed do the friction on the ice.

So even after being put on the icy surface, the speed in the box continues to be 𝑣 5.0 metres per second. Therefore, the momentum of the box when it’s on the floor is also equal to 25 kilograms metres per second. This is because the box loses no energy, sliding across the frictionless surface.