Understanding Kinetic Energy and Momentum in Inelastic Collisions: A Practical Example

In the study of physics, understanding the principles of momentum and kinetic energy is crucial. This article will explore how to apply these principles to solve a practical problem involving an inelastic collision. Specifically, we will use the example of two cars colliding to demonstrate the application of the conservation of momentum in such scenarios.

Introduction to Momentum and Inelastic Collisions

Momentum is a measure of the 'motion' of an object, and it is defined as the product of an object's mass and its velocity. In an inelastic collision, the colliding objects stick together after the collision, resulting in kinetic energy being converted into other forms of energy (such as heat and sound).

Problem Statement:

A 2-kg car moving towards the right at 4 m/s collides head-on with an 8-kg car moving towards the left at 2 m/s and they stick together. After the collision, what is the velocity of the combined bodies?

Solution to the Problem

Step 1: Calculate Initial Momentum

First, let's calculate the initial momentum of each car. Momentum is given by the formula:

momentum (p) mass (m) times; velocity (v)

1. Momentum of the 2-kg car moving to the right:

p_1 m_1 times; v_1 2 kg times; 4 m/s 8 kg m/s

2. Momentum of the 8-kg car moving to the left:

Since the left direction is typically considered negative:

p_2 m_2 times; v_2 8 kg times; -2 m/s -16 kg m/s

Step 2: Calculate Total Initial Momentum

The total initial momentum is the sum of the momentum of both cars:

momentum_{total initial} p_1 p_2

momentum_{total initial} 8 kg m/s (-16 kg m/s) -8 kg m/s

Step 3: Calculate Total Mass After Collision

The total mass of the combined bodies after the collision is the sum of the masses of the two cars:

mass_{total} m_1 m_2

mass_{total} 2 kg 8 kg 10 kg

Step 4: Use Conservation of Momentum to Find Final Velocity

According to the conservation of momentum, the total initial momentum must equal the total final momentum:

momentum_{total initial} momentum_{total final}

-8 kg m/s mass_{total} times; v_f

-8 kg m/s 10 kg times; v_f

Solving for the final velocity:

v_f frac{-8 kg cdot m/s}{10 kg} -0.8 m/s

This means the combined bodies are moving to the left at a velocity of -0.8 m/s.

Another Example: Inelastic Collision of Multiple Masses

Consider another scenario involving an inelastic collision of two masses. Let's say Massa has a mass of 40 g (.04 kg) moving at 9 m/s to the right, and Massb has a mass of 55 g (.055 kg) moving towards the left at 6 m/s. The collision is again inelastic, hence only momentum is conserved.

Let: mass_a m_a, v_a 9 m/s, mass_b m_b, v_b -6 m/s

The final velocity of the combined bodies after the collision can be calculated as:

v_2 frac{m_a cdot v_a m_b cdot v_b}{m_a m_b} frac{0.04 kg cdot 9 m/s 0.055 kg cdot (-6) m/s}{0.04 kg 0.055 kg}

v_2 frac{0.36 - 0.33}{0.095} frac{0.03}{0.095} 0.3157894737 m/s

Aproximately, the final velocity is 0.3158 m/s to the right.

This example further illustrates how the conservation of momentum can be applied to solve inelastic collision problems.