Question

A 3.0 -kg ball of clay with a speed of $21\mathrm{m}/\mathrm{s}$ is thrown against a wall and sticks to the wall. What is the magnitude of the impulse exerted on the ball?

Short answer

Answer: The magnitude of the impulse exerted on the ball is 63 kg m/s.

Step-by-step solution

Identify the given quantities

In this problem, we are given the mass (m) of the ball of clay as 3.0 kg and its initial velocity (v_initial) as 21 m/s. The ball sticks to the wall, which means that its final velocity (v_final) will be 0 m/s.

Calculate the initial momentum of the ball

The momentum is given by mass times velocity: $momentum=m\times v$. To find the initial momentum, we plug in the values for mass and initial velocity: $momentu{m}_{initial}=m\times v_{initial}$ $momentu{m}_{initial}=3.0kg\times 21m/s$ $momentu{m}_{initial}=63kgm/s$

Calculate the final momentum of the ball

As the ball sticks to the wall, its final velocity is 0 m/s. Therefore, the final momentum will also be 0: $momentu{m}_{final}=m\times v_{final}$ $momentu{m}_{final}=3.0kg\times 0m/s$ $momentu{m}_{final}=0kgm/s$

Calculate the change in momentum

The change in momentum (or impulse) is the difference between the final and initial momentums: $I=momentu{m}_{final}-momentu{m}_{initial}$ $I=0kgm/s-63kgm/s$ $I=-63kgm/s$

Find the magnitude of the impulse

The magnitude of the impulse is the absolute value of the impulse: $|I|=|-63kgm/s|$ $|I|=63kgm/s$ The magnitude of the impulse exerted on the ball is 63 kg m/s.

Key concepts

Conservation of Momentum

Conservation of momentum is a fundamental principle in physics stating that the total momentum of a closed system remains constant, provided no external forces act on it. In simpler terms, it's like ensuring that the account balance is unchanged, no matter how money gets transferred among accounts, as long as there is no deposit or withdrawal.
In our exercise, the ball of clay initially has momentum due to its motion towards the wall. When the ball hits the wall and sticks, an external force applies a change to its momentum. Thus, the total system here isn't entirely closed because the wall exerts an external force to stop the ball, affecting its momentum.
While the concept generally applies to isolated systems, understanding the interaction between the ball and the wall helps in visualizing real-world applications of momentum conservation.

Impulse-Momentum Theorem

The impulse-momentum theorem is a crucial concept in dynamics, relating the impulse applied to an object to its change in momentum. Impulse is essentially a measure of how much the momentum changes and is calculated by the product of force and the time during which the force acts.

• Impulse: Change in momentum
• Mathematically: \ I = \Delta p
• $\mathrm{\Delta }p=p_{final}-p_{initial}$

In our clay exercise, the impulse exerted by the wall on the ball is equal to the negative change in the ball's momentum. The negative sign indicates that the ball's direction changed as it goes from moving towards the wall to stopping.
Understanding this relationship allows you to see how force over time can drastically affect how objects speed up or slow down.

Physics Problem Solving

When it comes to solving physics problems like the one in this exercise, it's all about breaking down the problem into small, manageable steps:

• Identify what is given and what needs to be found.
• Use logical reasoning and equations relevant to the problem.
• Perform calculations meticulously to avoid small mistakes.
• Interpret the result and ensure it makes sense within the problem context.

In the case of the clay ball impacting the wall, the process involves clearly outlining the given variables, calculating initial and final momentum, and then determining impulse. Each step leads logically to the next, ensuring a systematic approach that meets the desired outcome.
Thus, honing these skills can make seemingly intricate physics exercises more navigable and less daunting.

Collision Dynamics

Collision dynamics examines the forces and interactions occurring during collisions, focusing on momentum changes, impacts, and energy exchange. In our clay example, the ball's sudden stop upon reaching the wall illustrates an inelastic collision, where two objects stick together post-impact, and kinetic energy is not conserved.

• Momentum before impact: Ball has significant momentum.
• During impact: Wall exerts a force, changing momentum.
• After impact: Ball's momentum is zero, highlighting energy dissipation.

The study of different collision types clarifies how objects behave when they meet. Understanding these principles is fundamental to analyzing real-world interactions, from simple classroom physics problems to complex traffic accident reconstructive studies.