Problem 5 - Applied Mathematics

Question

A particle of mass \(2 \mathrm{~kg}\) moving with speed \(4 \mathrm{~ms}^{-1}\) is given a blow which changes the speed to \(1 \mathrm{~ms}^{-1}\) without deflecting the particle from a straight line. The impulse of the blow is: (a) \(10 \mathrm{Ns}\), (b) \(6 \mathrm{Ns}\), (c) we do not know whether it is \(10 \mathrm{Ns}\) or \(6 \mathrm{Ns}\).

Step-by-Step Solution

Answer

The impulse of the blow is 6 Ns, so option (b) is correct.

Step 1: Understand the Impulse Formula

Impulse can be calculated using the formula: \[ J = \text{{change in momentum}} = m \times \text{{change in velocity}} \] where \( J \) is the impulse, \( m \) is the mass of the particle, and the change in velocity is the difference between the final and initial velocities.

Step 2: Identify Given Values

From the problem statement, identify the given values: Mass of the particle, \( m = 2 \text{ kg} \), Initial speed, \( u = 4 \text{ ms}^{-1} \), Final speed, \( v = 1 \text{ ms}^{-1} \).

Step 3: Calculate the Change in Velocity

Find the change in velocity: \[ \text{Change in velocity} = v - u = 1 \text{ ms}^{-1} - 4 \text{ ms}^{-1} = -3 \text{ ms}^{-1} \]

Step 4: Calculate the Impulse

Substitute the values into the impulse formula: \[ J = m \times \text{change in velocity} = 2 \text{ kg} \times (-3 \text{ ms}^{-1}) = -6 \text{ Ns} \] The negative sign indicates the direction of the impulse is opposite to the direction of the initial velocity.

Step 5: Determine the Magnitude of the Impulse

The magnitude of the impulse is: \[ |J| = 6 \text{ Ns} \]

Step 6: Choose the Correct Option

The correct option from the given choices is (b) \(6 \text{ Ns}\).

Key Concepts

Change in Momentum

In physics, momentum is a crucial concept that combines mass and velocity. It is expressed by the equation: \( p = m \times v \), where \( p \) stands for momentum, \( m \) for mass, and \( v \) for velocity. When an object’s velocity changes, its momentum changes. This change in momentum is what we refer to when we say an impulse has occurred. Impulse is the effect of a force acting over a specific period of time to change the object's momentum. For better understanding, think of it as a force that helps in accelerating or decelerating an object, resulting in a change in its velocity. This can be seen clearly in everyday events such as kicking a soccer ball, where the force from your foot changes the ball's momentum.

Impulse Formula

When calculating impulse, we use the formula: \( J = \text{{change in momentum}} = m \times \text{{change in velocity}} \). Here, \( J \) represents the impulse, and it’s calculated by multiplying the mass of the object by the change in its velocity. Impulse essentially measures the overall impact of a force applied over a time period. For instance, a soccer player kicking a ball exerts a force that changes the ball’s velocity, thereby changing its momentum. This change, whether it increases or decreases, denotes the impulse. Additionally, impulse is not only about the amount of force but how long it acts upon the object. Therefore, even a small force applied over a longer period can produce a significant impulse.

Velocity Change

Velocity is the speed of an object in a given direction. Any modification in this speed or direction is termed as a change in velocity. In our exercise, the particle’s speed changes from \( 4 \text{ ms}^{-1} \) to \( 1 \text{ ms}^{-1} \). To calculate this change, we subtract the initial velocity \( u \) from the final velocity \( v \): \( \text{Change in velocity} = v - u \). Hence, \( \text{Change in velocity} = 1 \text{ ms}^{-1} - 4 \text{ ms}^{-1} = -3 \text{ ms}^{-1} \). The negative sign indicates a decrease in speed, showing the direction of change is opposite to the initial movement. Understanding the change in velocity helps us understand how forces affect motion, which is foundational in studying mechanics.

Mass and Velocity

Mass and velocity are the two components determining an object's momentum. Mass is the amount of matter contained in an object and remains constant unless the object is structurally altered. Velocity, as discussed, is the speed and direction of an object's movement. When analyzing how forces affect objects, it's essential to consider both mass and velocity. For instance, in the exercise, the particle's mass is given as \( 2 \text{ kg} \), and its velocity changes from \( 4 \text{ ms}^{-1} \) to \( 1 \text{ ms}^{-1} \). Understanding how these two factors interact helps us calculate changes in momentum and impulse, as seen in the formula: \( J = m \times \text{{change in velocity}} \). This relationship showcases how different forces impact various objects based on their mass and velocity.