Problem 24 - College Physics

Question

This is a symbolic version of Problem 23. A girl of mass \(m_{G}\) is standing on a plank of mass \(m_{p}\). Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity \(x_{C P}\) to the right relative to the plank. (The subscript \(G P\) denotes the girl relative to plank.) (a) What is the velocity \(v_{P}\) of the plank relative to the surface of the ice? (b) What is the girl's velocity \(v_{C I}\) relative to the ice surface?

Step-by-Step Solution

Answer

The velocity of the plank relative to ice is \(v_P = - \frac{m_G * v_{GP}}{m_P}\) and the girl's velocity relative to ice surface is \(v_{CI} = v_{GP} * (1-\frac{m_G}{m_P})\).

Step 1: Comprehend the Problem

The girl and the plank are initially at rest on a frictionless ice surface. When the girl moves relative to the plank, the momentum of the system (girl + plank) should be conserved because there are no external forces acting on it. Thus, the sum of the momenta of the girl and plank before and after the girl starts moving is equal. The momentum of any object is given by the product of its mass and velocity.

Step 2: Calculate Velocity of the Plank - Step 1

According to the conservation of momentum, the initial momentum (when both girl and plank are at rest) of the system is equal to the final momentum of the system (when the girl is walking on the plank). The initial momentum is zero because both the plank and the girl are at rest. If we let the direction the girl walks as positive, then we can write the momentum conservation equation as follows: \(m_G * v_{GP} = m_P * (-v_P)\). Here, \(v_P\) is negative because the plank will move in the opposite direction to conserve momentum.

Step 3: Calculate Velocity of the Plank - Step 2

Rearranging the above equation, we can find the velocity of the plank relative to ice: \(v_P = - \frac{m_G * v_{GP}}{m_P}\).

Step 4: Calculate Girl's Velocity Relative to Ice - Step 1

The velocity of the girl relative to the ice (\(v_{CI}\)) is the summation of girl’s velocity relative to the plank (\(v_{GP}\)) and plank’s velocity relative to the ice (\(v_P\)): \(v_{CI}= v_{GP} + v_P\).

Step 5: Calculate Girl's Velocity Relative to Ice - Step 2

Substitute the previously calculated \(v_P\) value in the above equation: \(v_{CI} = v_{GP} - \frac{m_G * v_{GP}}{m_P}\). After simplifying, the resultant velocity of the girl with respect to ice is found to be: \(v_{CI} = v_{GP} * (1-\frac{m_G}{m_P})\).

Key Concepts

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion an object possesses. Simply put, momentum is the product of an object's mass and its velocity, mathematically expressed as p = mv, where p represents momentum, m is the mass, and v is the velocity.

The importance of momentum becomes particularly evident when discussing the principle of conservation of momentum. According to this principle, in a closed system where no external forces are acting, the total momentum remains constant over time. This is central to understanding interactions such as collisions or, as in this exercise, the movement of a girl on a plank on a frictionless surface. When the girl starts walking, she generates momentum in one direction, while the plank moves in the opposite direction with an equal amount of momentum, thus conserving the momentum of the system as a whole.

The step-by-step solution shown in the original exercise applies the principle by setting the total momentum before movement (which is zero, as both the girl and the plank are at rest) equal to the total momentum after the girl starts walking.

Frictionless Surface

A frictionless surface is an idealized concept used in physics to simplify the study of motion without the complicating factor of friction. In reality, all surfaces exert some form of friction, which opposes the motion of objects. However, in theoretical models, a frictionless surface allows for the perfect conservation of momentum as there are no external forces acting to slow down or stop the moving objects.

In the context of the original exercise, the frozen lake is considered a frictionless flat surface. This assumption ensures that the only factors affecting the motion of the girl and the plank are their interactions with each other. The absence of friction is why the plank can move in the opposite direction to that of the girl without any hindrance. If friction were present, the force it exerted would have to be considered when calculating the velocities and momentum, complicating the problem significantly.

Relative Velocity

Relative velocity is the velocity of an object as observed from the frame of reference of another moving object. It is a vector quantity, which means it has both magnitude and direction. Relative velocity comes into play whenever you are considering the motion of one object as seen from another object that is also in motion.

In the exercise, the velocity of the girl relative to the plank, denoted as v_GP, is her velocity as seen from the plank's reference frame. Conversely, v_CI, the girl's velocity related to the ice, is observed from the stationary frame of the ice surface. The solution method involves understanding that the girl's relative velocity concerning the ice is the combination of her velocity relative to the plank and the plank's velocity relative to the ice. To find the true velocity of the girl with respect to the stationary surface of the ice, one must sum her velocity relative to the moving plank with the velocity of the plank itself, as they contribute to her overall motion in the reference frame of an outside observer on the ice.