A Rubber Ball Of Mass 415 G Falls Vertically 160 Cm Onto A Section Of Horizontal Ground. Given That The Ball Always Rebounds To One-quarter Of The Height From Which It Fell, What Is The Magnitude Of The Change In Momentum As A Result Of The Second

by ADMIN 248 views

Introduction

When an object falls from a certain height and rebounds back up, it undergoes a change in momentum. The change in momentum is a result of the force exerted on the object during the collision with the ground. In this scenario, a rubber ball of mass 415 g falls vertically 160 cm onto a section of horizontal ground. Given that the ball always rebounds to one-quarter of the height from which it fell, we need to determine the magnitude of the change in momentum as a result of the second rebound.

Understanding the Problem

To solve this problem, we need to understand the concept of momentum and how it changes when an object falls and rebounds. Momentum is the product of an object's mass and velocity. When the ball falls, its velocity increases due to the acceleration caused by gravity. As it hits the ground, its velocity is reversed, and it rebounds back up. The change in momentum is a result of the force exerted on the ball during the collision with the ground.

Calculating the Initial Velocity

First, we need to calculate the initial velocity of the ball when it falls from a height of 160 cm. We can use the equation for free fall:

v² = 2gh

where v is the final velocity, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the ball falls.

import math

m = 415  # mass in grams
h = 160  # height in cm
g = 9.8  # acceleration due to gravity in m/s²



h_m = h / 100



v_initial = math.sqrt(2 * g * h_m)


print("Initial velocity:", v_initial, "m/s")

Calculating the Change in Momentum

The change in momentum is a result of the force exerted on the ball during the collision with the ground. We can use the equation for impulse:

Δp = F × Δt

where Δp is the change in momentum, F is the force exerted on the ball, and Δt is the time over which the force is applied.

However, we don't have the force or time values. Instead, we can use the equation for momentum:

p = m × v

We can calculate the change in momentum by finding the difference between the final and initial momenta.

Calculating the Final Velocity

After the ball rebounds, its velocity is reversed, and it reaches a new height of one-quarter of the original height. We can use the equation for free fall again to calculate the final velocity:

v² = 2gh

# Calculate final velocity
v_final = math.sqrt(2 * g * (h_m / 4))

print("Final velocity:", v_final, "m/s")

Calculating the Change in Momentum

Now that we have the initial and final velocities, we can calculate the change in momentum:

Δp = m × (v_final - v_initial)

# Calculate change in momentum
delta_p = m * (v_final - v_initial)

print("Change in momentum:", delta_p, "kg m/s")

Conclusion

In this scenario, a rubber ball of mass 415 g falls vertically 160 cm onto a section of horizontal ground. Given that the ball always rebounds to one-quarter of the height from which it fell, we calculated the magnitude of the change in momentum as a result of the second rebound. The change in momentum is a result of the force exerted on the ball during the collision with the ground.

Discussion

The change in momentum is an important concept in physics, and it has many real-world applications. In this scenario, we used the equation for impulse to calculate the change in momentum. However, we didn't have the force or time values, so we used the equation for momentum instead.

The change in momentum is a result of the force exerted on the ball during the collision with the ground. The force exerted on the ball is a result of the normal force exerted by the ground on the ball. The normal force is a result of the ball's weight and the ground's reaction force.

The change in momentum is an important concept in physics, and it has many real-world applications. In this scenario, we used the equation for impulse to calculate the change in momentum. However, we didn't have the force or time values, so we used the equation for momentum instead.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Code

import math


m = 415  # mass in grams
h = 160  # height in cm
g = 9.8  # acceleration due to gravity in m/s²



h_m = h / 100



v_initial = math.sqrt(2 * g * h_m)



v_final = math.sqrt(2 * g * (h_m / 4))



delta_p = m * (v_final - v_initial)


print("Initial velocity:", v_initial, "m/s")
print("Final velocity:", v_final, "m/s")
print("Change in momentum:", delta_p, "kg m/s")


# A rubber ball of mass 415 g falls vertically 160 cm onto a section of horizontal ground

## Q&A: Understanding the Change in Momentum

### Q: What is the change in momentum of the rubber ball after it rebounds from the ground?

A: The change in momentum of the rubber ball after it rebounds from the ground is a result of the force exerted on the ball during the collision with the ground. We can calculate the change in momentum by finding the difference between the final and initial momenta.

### Q: How do we calculate the initial velocity of the rubber ball?

A: We can use the equation for free fall to calculate the initial velocity of the rubber ball:

v² = 2gh

where v is the final velocity, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the ball falls.

### Q: What is the final velocity of the rubber ball after it rebounds from the ground?

A: After the rubber ball rebounds, its velocity is reversed, and it reaches a new height of one-quarter of the original height. We can use the equation for free fall again to calculate the final velocity:

v² = 2gh

### Q: How do we calculate the change in momentum of the rubber ball?

A: We can calculate the change in momentum by finding the difference between the final and initial momenta:

Δp = m × (v_final - v_initial)

where m is the mass of the rubber ball, v_final is the final velocity, and v_initial is the initial velocity.

### Q: What is the significance of the change in momentum of the rubber ball?

A: The change in momentum of the rubber ball is an important concept in physics, and it has many real-world applications. In this scenario, we used the equation for impulse to calculate the change in momentum. However, we didn't have the force or time values, so we used the equation for momentum instead.

### Q: What are some real-world applications of the change in momentum?

A: The change in momentum has many real-world applications, including:

* Calculating the force exerted on an object during a collision
* Determining the energy transferred during a collision
* Understanding the behavior of objects in motion

### Q: How do we ensure that the change in momentum is calculated accurately?

A: To ensure that the change in momentum is calculated accurately, we need to:

* Use the correct equations and formulas
* Ensure that the values used are accurate and consistent
* Consider the units and dimensions of the values used

### Q: What are some common mistakes to avoid when calculating the change in momentum?

A: Some common mistakes to avoid when calculating the change in momentum include:

* Failing to consider the units and dimensions of the values used
* Using incorrect or inconsistent values
* Failing to account for the force exerted on the object during the collision

### Q: How do we apply the concept of change in momentum to real-world problems?

A: We can apply the concept of change in momentum to real-world problems by:

* Calculating the force exerted on an object during a collision
* Determining the energy transferred during a collision
* Understanding the behavior of objects in motion

### Q: What are some additional resources for learning more about the change in momentum?

A: Some additional resources for learning more about the change in momentum include:

* Textbooks on physics and mechanics
* Online tutorials and videos
* Scientific articles and research papers

## Conclusion

The change in momentum is an important concept in physics, and it has many real-world applications. By understanding the change in momentum, we can calculate the force exerted on an object during a collision, determine the energy transferred during a collision, and understand the behavior of objects in motion.</code></pre>