How to Find Final Momentum After Collision: A Comprehensive Guide

When two objects collide, the final momentum of the system is a crucial quantity to determine. This comprehensive guide will walk you through the step-by-step process of finding the final momentum after a collision using the Impulse-Momentum Theorem and the Conservation of Linear Momentum. We’ll cover various examples, including 2D collisions and inelastic collisions, to provide you with a thorough understanding of this fundamental concept in physics.

Impulse-Momentum Theorem

The Impulse-Momentum Theorem is a powerful tool for calculating the final momentum after a collision. It states that the change in momentum of an object is equal to the impulse applied to the object. Mathematically, this can be expressed as:

$\Delta p = F \cdot \Delta t$

Where:
– $\Delta p$ is the change in momentum
– $F$ is the force applied
– $\Delta t$ is the time interval over which the force is applied

To find the final momentum using the Impulse-Momentum Theorem, follow these steps:

1. Determine the initial momentum and direction:
2. Calculate the initial momentum using the formula: $p_i = m \cdot v_i$, where $m$ is the mass and $v_i$ is the initial velocity.

3. Identify the direction of the initial momentum.

4. Identify the impulse and its direction:

5. Calculate the impulse using the formula: $\Delta p = F \cdot \Delta t$, where $F$ is the force and $\Delta t$ is the time interval over which the force is applied.

6. Identify the direction of the impulse.

7. Use the Impulse-Momentum Theorem to find the final momentum:

8. Plug the initial momentum and impulse into the equation: $p_i + \Delta p = p_f$, where $p_f$ is the final momentum.
9. Solve for $p_f$ to find the final momentum.

Example Problem 1 – Impulse-Momentum Theorem

A tennis ball with an initial momentum of 2 kgm/s to the right feels an impulse of 5 kgm/s to the left from a tennis racket. What is its final momentum?

1. Determine the initial momentum and direction:

2. $p_i = 2$ kgm/s to the right.

3. Identify the impulse and its direction:

4. $\Delta p = -5$ kgm/s to the left.

5. Use the Impulse-Momentum Theorem to find the final momentum:

6. $p_i + \Delta p = p_f$
7. $2$ kgm/s – $5$ kgm/s = $p_f$
8. $p_f = -3$ kgm/s to the left.

Conservation of Linear Momentum

The Conservation of Linear Momentum is another fundamental principle that can be used to find the final momentum after a collision. It states that the total momentum of a closed system is conserved, meaning that the initial momentum is equal to the final momentum.

To find the final velocity in a 2D collision using the Conservation of Momentum, follow these steps:

1. Identify the mass of each object, their velocities, and their angle of trajectory after the collision:

2. Identify the masses, initial velocities, and final angles of each object.

3. Break down the total momentum into the directional components and plug in the known values:

4. For momentum in the X direction: $m_1v_{1x} + m_2v_{2x} = m_1v_{1}^{‘}cos(\theta_1) + m_2v_{2}^{‘}cos(\theta_2)$

5. For momentum in the Y direction: $m_1v_{1y} + m_2v_{2y} = m_1v_{1}^{‘}sin(\theta_1) + m_2v_{2}^{‘}sin(\theta_2)$.

6. Choose one of the component momentum directions to solve for the unknown velocity:

7. Select one of the directional components to solve for the unknown velocity.

8. Verify that the answer for the unknown velocity is correct by plugging into the other directional component equation for momentum and that the initial momentum is equal to the final momentum:

9. Verify the solution by plugging it into the other directional component equation and ensuring that the initial momentum equals the final momentum.

Example Problem 2 – 2 Dimensional Conservation of Momentum

Two blocks are sliding on a frictionless surface. Block A has a mass of 2 kg and is sliding upward at 4.5 m/s. Block B has a mass of 4 kg and is sliding to the right at 3 m/s. After the collision, block A is traveling at an unknown velocity at an angle of 45 degrees from the X-axis, block B is traveling at 2 m/s at an angle of 30 degrees from the X-axis. What is the final velocity of block A?

1. Identify the mass of each object, their velocities, and their angle of trajectory after the collision:
2. Block A: $m = 2$ kg, $v_i = 4.5$ m/s, $\theta = 45^{\circ}$

3. Block B: $m = 4$ kg, $v_i = 3$ m/s, $v_f = 2$ m/s, $\theta = 30^{\circ}$.

4. Break down the total momentum into the directional components and plug in the known values:

5. Momentum in the X direction: $(2$ kg$)(0$ m/s$) + (4$ kg$)(3$ m/s$) = (2$ kg$)v_{1}^{‘}cos(45) + (4$ kg$)(2$ m/s$)cos(30)$

6. Momentum in the Y direction: $(2$ kg$)(4.5$ m/s$) + (4$ kg$)(0$ m/s$) = (2$ kg$)v_{1}^{‘}sin(45) + (4$ kg$)(2$ m/s$)sin(30)$.

7. Choose one of the component momentum directions to solve for the unknown velocity:

8. Solve for $v_{1}^{‘}$ using either the X or Y direction momentum equation.

9. Verify that the answer for the unknown velocity is correct by plugging into the other directional component equation for momentum and that the initial momentum is equal to the final momentum:

10. Verify the solution by plugging it into the other directional component equation and ensuring that the initial momentum equals the final momentum.

Example Problem – Inelastic Collision

Two train cars, one with a mass of 90,000 kg and the other with a mass of 90,000 kg, collide and couple together. If the initial momentum is 180,000 kgm/s, what is the final velocity of the coupled train cars?

1. Determine the initial momentum and direction:

2. $p_i = 180,000$ kgm/s.

3. Identify the impulse and its direction:

4. Since the cars couple together, the final momentum is the sum of the masses times the final velocity: $p_f = (90,000$ kg + $90,000$ kg$) \cdot v_f$.

5. Use the Impulse-Momentum Theorem to find the final momentum:

6. $p_i = p_f$
7. $180,000$ kgm/s = $180,000$ kg $\cdot v_f$
8. $v_f = 2$ m/s.

Additional Considerations

• In an elastic collision, the total kinetic energy of the system is conserved, in addition to the conservation of momentum. This can provide additional constraints and equations to solve for the final velocities.
• Inelastic collisions involve a loss of kinetic energy, and the final velocities are typically lower than the initial velocities. The Impulse-Momentum Theorem is particularly useful in these cases.
• For more complex collision scenarios, such as those involving multiple objects or rotational motion, additional principles and equations may be required to fully analyze the problem.

References

1. Study.com. (2021). How to Use the Impulse-Momentum Theorem to Calculate a Final Momentum. Retrieved from https://study.com/skill/learn/how-to-use-the-impulse-momentum-theorem-to-calculate-a-final-momentum-explanation.html
2. JH Scientific. (2008). Conservation of Linear Momentum & Occupant Kinematics. Retrieved from https://www.jhscientific.com/downloads/COLMandOccupantKinematics.pdf
3. Tennessee Technological University. (n.d.). Chapter 7 Linear Momentum and Collisions. Retrieved from https://www2.tntech.edu/leap/murdock/books/v1chap7.pdf
4. Study.com. (2021). Using the Conservation of Momentum to Find a Final Velocity in 2 Dimensions. Retrieved from https://study.com/skill/learn/using-the-conservation-of-momentum-to-find-a-final-velocity-in-2-dimensions-explanation.html
5. YouTube. (2021). How Do You Calculate Final Velocity in Inelastic Collisions? Retrieved from https://www.youtube.com/watch?v=dGHeVjT1xhQ