Problem 23 - College Physics

Question

A \(45.0-\mathrm{kg}\) girl is standing on a \(150-\mathrm{kg}\) 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 \(1.50 \mathrm{~m} / \mathrm{s}\) to the right relative to the plank. (a) What is her velocity relative to the surface of the ice? (b) What is the velocity of the plank relative to the surface of the ice?

Step-by-Step Solution

Answer

The girl's velocity relative to the ice is \(1.50 \mathrm{m/s}\) to the right. The velocity of the plank relative to the ice is \( -0.45 \mathrm{m/s}\) to the left.

Step 1: Compute total initial momentum

First it's important understand that the initial total momentum equals to the final total momentum because of conservation of momentum principle in an isolated system. The initial total momentum is 0 because both the plank and the girl are at rest.

Step 2: Calculate girl's velocity relative to the ice

The girl is moving right relative to the plank at a speed of \(1.50 \mathrm{m/s}\). Therefore, her velocity relative to the ice is the same \(1.50 \mathrm{m/s}\) to the right. This is because the plank is at rest relative to the ice.

Step 3: Calculate velocity of the plank

The girl and the plank together form an isolated system (there's no external force). Hence, the total momentum must remain 0 (from step1). The plank must move so that its momentum cancels out the girl's momentum. So, \(m_{g}v_{g} + m_{p}v_{p} = 0\), where: \(m_{g}\) = mass of the girl = 45.0 kg, \(v_{g}\) = velocity of the girl = 1.50 m/s; \(m_{p}\) = mass of the plank = 150 kg; \(v_{p}\) = velocity of the plank (the velocity we are looking for). Rearranging the equation for \(v_{p}\), we have \(v_{p} = -\frac{m_{g}v_{g}}{m_{p}}\) where negative sign suggests the plank will move to the left (opposite to the motion of girl). By substituting the given values, velocity of plank can be calculated.

Key Concepts

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion an object has. It can be thought of as the 'oomph' that keeps an object moving. Mathematically, momentum (\textbf{p}) is defined as the product of an object's mass (\textbf{m}) and its velocity (\textbf{v}). The equation for linear momentum is given by \( p = m \times v \).

When objects interact, such as colliding or exerting forces on each other, their total momentum before the interaction is equal to their total momentum afterward, provided they form an isolated system. This principle is known as the conservation of momentum. It is particularly useful in solving problems where two or more objects interact on a frictionless surface, just like the girl and the plank on the ice in our exercise.

Isolated System

In physics, an isolated system refers to a collection of objects that do not experience any external forces. No matter, energy, or forces enter or leave the system, making it closed off from its surroundings. This concept is crucial for the conservation of momentum because, in an isolated system, the total momentum remains constant over time.

An important implication of being 'isolated' is that internal forces within the system, such as the girl walking on the plank, do not affect the total momentum of the system. This aspect was vital in solving the exercise, as it allowed us to say that despite the girl's motion, the total momentum of the girl-plank system remained zero before and after she started walking.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. It is an essential concept when addressing problems involving motion because how fast one object is moving relative to another can differ from how fast it is moving relative to a stationary observer. In terms of formula, if object A has velocity \( v_A \) and object B has velocity \( v_B \), then the velocity of A relative to B is \( v_{AB} = v_A - v_B \), and vice versa.

In the context of the exercise, the girl's velocity relative to the ice was the same as her velocity relative to the plank, because the plank was initially at rest relative to the ice. When she starts walking at \(1.50 \mathrm{m/s}\) relative to the plank, from an observer on the ice, she is also moving at \(1.50 \mathrm{m/s}\) to the right.

Frictionless Surface

A frictionless surface is idealized in physics to describe a surface over which there is no frictional force resisting the motion of an object. This simplification allows us to ignore energy losses and forces that would otherwise impact the momentum of moving objects. In real life, no surface is completely frictionless; however, surfaces like ice can be close to this ideal, making them excellent examples for such problems.

In our problem, the ice provides a nearly frictionless surface, which is why the only forces acting on the girl and plank system are internal (the girl walking). Therefore, external forces like friction do not interfere with the conservation of momentum, allowing us to apply that principle to predict the motion of both the girl and the plank after the girl begins walking.