Problem 17 - Applied Mathematics

Question

A ball moving on a horizontal floor hits a smooth vertical wall normally. Calculate the speed with which it leaves the wall if: (a) the speed when approaching the wall is \(3 \mathrm{~ms}^{-1}\), (b) the coefficient of restitution is \(\frac{1}{2}\), (c) the mass of the ball is \(0.4 \mathrm{~kg}\).

Step-by-Step Solution

Answer

The speed with which the ball leaves the wall is 1.5 ms^{-1}.

Step 1: Identify given values

Let's identify the given values from the problem:(a) Initial speed approaching the wall: \( u = 3 \text{ ms}^{-1} \)(b) Coefficient of restitution: \( e = \frac{1}{2} \)(c) Mass of the ball: \( m = 0.4 \text{ kg} \)

Step 2: Understand the coefficient of restitution

The coefficient of restitution \( e \) is a measure of how much kinetic energy is conserved in a collision. It is defined by the formula: \[ e = \frac{\text{Speed after collision}}{\text{Speed before collision}} \]For our problem: \( e = \frac{v}{u} \).

Step 3: Insert the given values into the coefficient of restitution formula

We know \( u = 3 \text{ ms}^{-1} \) and \( e = \frac{1}{2} \). Using the formula \( e = \frac{v}{u} \), we get: \[ \frac{1}{2} = \frac{v}{3} \]

Step 4: Solve for the final speed

Rearranging the equation for \( v \): \[ v = u \times e = 3 \times \frac{1}{2} = 1.5 \text{ ms}^{-1} \]Therefore, the speed of the ball after it leaves the wall is \(1.5 \text{ ms}^{-1} \).

Key Concepts

Elastic Collision

An elastic collision is a type of collision in which the total kinetic energy of the colliding bodies is conserved. This means that the sum of the kinetic energies before and after the collision remains the same. In our problem, the ball hitting the wall is an example of a collision where we use the coefficient of restitution to understand how much kinetic energy is conserved in this event. Let's look at the concept of the coefficient of restitution more closely.

The coefficient of restitution, denoted as 'e', measures the ratio of the relative speeds of separation to the relative speeds of approach. Mathematically, it is defined as: \[ e = \frac{\text{Speed after collision}}{\text{Speed before collision}} \]

For our practice problem, with an initial speed of 3 ms\text{-1} and a coefficient of restitution of \(\frac{1}{2}\), applying the formula helps us find that the speed after the collision is reduced to 1.5 ms\text{-1}. This reduction reflects the decrease in kinetic energy, indicating that the collision is not perfectly elastic.

Physics Problem

Physics problems like the one presented here are real-world scenarios that can be broken down using principles and laws of physics. In this case, we deal with a problem involving collisions, kinetic energy conservation, and speed calculations. Physics allows us to predict the outcome of such collisions by knowing only a few key values: the initial speed, the coefficient of restitution, and other properties like mass.

After identifying the relevant values from the problem—initial speed of 3 ms\text{-1}, a coefficient of restitution of \(\frac{1}{2}\), and the mass of the ball being 0.4 kg—we apply these values to the theoretical formulas to find the desired quantity. This iterative approach in breaking down the complexity and solving for each quantity step by step develops an understanding of the problem and applies theoretical knowledge to practical situations.

Velocity Calculation

In physics, calculating velocities after collisions is crucial for understanding the dynamics of moving objects. Here’s a structured way to approach these calculations:

• Identify the initial conditions, such as the initial speed and the coefficient of restitution.
• Apply the relevant formulas, such as the coefficient of restitution formula.
• Insert the given values into the formula and solve for the final velocity.

For the presented problem, we derived the final velocity step-by-step as follows:
The initial speed approaching the wall was given as 3 ms\text{-1}.
The coefficient of restitution was given as \(\frac{1}{2}\).
By using the formula
\[ e = \frac{v}{u}\]
and substituting the known values, we get:
\[ \frac{1}{2} = \frac{v}{3}\]
Rearranging and solving, we find:
\[ v = u \times e = 3 \times \frac{1}{2} = 1.5 \text{ ms}^{-1} \]

This structured calculation ensures that we accurately determine the speed of the ball after it bounces off the wall, thereby solidifying our understanding of velocity changes during elastic collisions.