Is Impulse Conserved? A Comprehensive Guide

Summary

Impulse is a fundamental concept in physics that relates to the change in momentum of an object. The conservation of impulse is a critical principle in understanding various physical phenomena, particularly in collisions and interactions between objects. This comprehensive guide delves into the theoretical background, experimental evidence, and quantifiable data on the conservation of impulse.

Theoretical Background

Impulse and Momentum

Impulse is defined as the product of force and time, mathematically represented as:

$\text{Impulse} = F \times t$

where $F$ is the force acting on an object and $t$ is the time over which the force is applied. This impulse is directly related to the change in momentum of the object, as expressed by the equation:

$\text{Impulse} = \Delta p = m \times \Delta v$

where $m$ is the mass of the object, $\Delta v$ is the change in velocity, and $\Delta p$ is the change in momentum.

Conservation of Impulse

The conservation of impulse is a fundamental principle in physics, stating that the total impulse of a closed system remains constant over time. This means that the sum of the impulses of all objects in the system is conserved, provided there are no external forces acting on the system. Mathematically, this can be represented as:

$\sum \text{Impulse}_i = \sum F_i \times t_i = \text{constant}$

where $i$ represents each object in the system.

Experimental Evidence

Several experiments have been conducted to demonstrate the conservation of impulse. These experiments have focused on both elastic and inelastic collisions.

Elastic Collisions

In elastic collisions, the total momentum before and after the collision remains the same. This is evident from the equation:

$p_{\text{before}} = p_{\text{after}}$

where $p_{\text{before}}$ and $p_{\text{after}}$ are the total momenta before and after the collision, respectively.

Inelastic Collisions

In inelastic collisions, the objects stick together after the collision. In this case, the total momentum before the collision is equal to the total momentum after the collision, but the kinetic energy is not conserved.

Quantifiable Data

To further illustrate the conservation of impulse, let’s consider some quantifiable data points from experiments.

Elastic Collision

1. Initial momentum of cart A: 10 kg m/s
2. Initial momentum of cart B: 0 kg m/s (at rest)
3. Final momentum of cart A: 5 kg m/s
4. Final momentum of cart B: 5 kg m/s
5. Total momentum before collision: 10 kg m/s
6. Total momentum after collision: 10 kg m/s

These data points demonstrate that the total impulse, and consequently the total momentum, remains conserved in an elastic collision.

Inelastic Collision

1. Initial momentum of cart A: 10 kg m/s
2. Initial momentum of cart B: 0 kg m/s (at rest)
3. Final momentum of the combined system: 10 kg m/s
4. Total momentum before collision: 10 kg m/s
5. Total momentum after collision: 10 kg m/s

In this inelastic collision, the total momentum before and after the collision is the same, even though the kinetic energy is not conserved.

Theorem and Formulas

1. Impulse-Momentum Theorem: The impulse of a force is equal to the change in momentum of the object on which the force acts.
   $\text{Impulse} = \Delta p$

2. Conservation of Momentum: In a closed system, the total momentum before a collision is equal to the total momentum after the collision.
   $p_{\text{before}} = p_{\text{after}}$

3. Impulse Formula: Impulse is the product of force and time.
   $\text{Impulse} = F \times t$

4. Momentum Formula: Momentum is the product of mass and velocity.
   $p = m \times v$

Examples and Numerical Problems

1. Example 1: A 2 kg object is moving at 5 m/s. It collides with a 3 kg object moving at 3 m/s in the opposite direction. Assuming an elastic collision, calculate the final velocities of the two objects.

Given:
– Mass of object 1: $m_1 = 2 \text{ kg}$
– Initial velocity of object 1: $v_1 = 5 \text{ m/s}$
– Mass of object 2: $m_2 = 3 \text{ kg}$
– Initial velocity of object 2: $v_2 = -3 \text{ m/s}$

Using the conservation of momentum:
$p_{\text{before}} = p_{\text{after}}$
$m_1 v_1 + m_2 v_2 = m_1 v_1′ + m_2 v_2’$
Solving for the final velocities:
$v_1′ = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} = \frac{2 \times 5 + 3 \times (-3)}{2 + 3} = 1 \text{ m/s}$
$v_2′ = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} = \frac{2 \times 5 + 3 \times (-3)}{2 + 3} = -1 \text{ m/s}$

1. Numerical Problem: A 5 kg object is moving at 10 m/s. It collides with a 3 kg object moving at 8 m/s in the opposite direction. Assuming an inelastic collision, calculate the final velocity of the combined system.

Given:
– Mass of object 1: $m_1 = 5 \text{ kg}$
– Initial velocity of object 1: $v_1 = 10 \text{ m/s}$
– Mass of object 2: $m_2 = 3 \text{ kg}$
– Initial velocity of object 2: $v_2 = -8 \text{ m/s}$

Using the conservation of momentum:
$p_{\text{before}} = p_{\text{after}}$
$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_{\text{final}}$
Solving for the final velocity:
$v_{\text{final}} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} = \frac{5 \times 10 + 3 \times (-8)}{5 + 3} = 4 \text{ m/s}$

Figures and Diagrams

Figure 1: Illustration of an elastic collision between two objects.

Figure 2: Illustration of an inelastic collision between two objects.

Conclusion

The conservation of impulse is a fundamental principle in physics that has been extensively studied and verified through various experiments. The data and examples presented in this guide demonstrate the quantifiable nature of impulse conservation, both in elastic and inelastic collisions. Understanding the conservation of impulse is crucial for analyzing and predicting the behavior of objects in various physical systems.

References

1. Ole Miss Physics. (n.d.). Experiment 5: Conservation of Momentum. Retrieved from https://www.phy.olemiss.edu/~thomas/weblab/107_webpage_upload/1_107_COVID_web_items/107_Conservation%20of%20Momentum/107_Conservation%20of%20momentum_procedure_COVID.pdf
2. YouTube. (2022). Impulse and the Conservation of Momentum – Chapter 7. Retrieved from https://www.youtube.com/watch?v=W-xhOh0_cTs
3. The Physics Classroom. (n.d.). Momentum Change and Impulse Connection. Retrieved from https://www.physicsclassroom.com/class/momentum/Lesson-1/Momentum-and-Impulse-Connection
4. Paulding County School District. (n.d.). AP Physics 1 Investigation 5: Impulse and Momentum. Retrieved from https://www.paulding.k12.ga.us/cms/lib010/GA01903603/Centricity/Domain/525/ap%20physics%201investigation5impulseandmomentum.pdf