How to Find Energy Lost in a Collision: A Comprehensive Guide

When two objects collide, energy is transferred between them. In some cases, this energy is lost as it is converted into other forms like heat or sound. The process of determining the amount of energy lost in a collision is crucial in various fields, such as physics, engineering, and sports science. In this blog post, we will explore how to find the energy lost in a collision, including the formulas and mathematical expressions involved. Let’s dive in!

How to Calculate Energy Lost in a Collision

how to find energy lost in a collision — Image by Lookang – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.

Identifying the Initial and Final Kinetic Energy

To calculate the energy lost in a collision, we need to start by identifying the initial and final kinetic energy of the objects involved.

Kinetic energy is the energy possessed by an object due to its motion and is given by the equation:

$K.E. = \frac{1}{2}mv^2$

Where:
– $K.E.$ represents the kinetic energy of the object,
– $m$ stands for the mass of the object, and
– $v$ represents the velocity of the object.

It’s important to note that the kinetic energy depends on both the mass and velocity of an object. By determining the initial and final kinetic energy values, we can calculate the energy lost in a collision.

The Mathematical Formula for Energy Loss

The energy lost in a collision can be calculated using the following formula:

$\text{Energy Loss} = \text{Initial Kinetic Energy} - \text{Final Kinetic Energy}$

Simply subtract the final kinetic energy from the initial kinetic energy to find the energy lost. This formula allows us to quantify the amount of energy dissipated during a collision.

Worked Out Example on Energy Loss Calculation

Let’s walk through a worked-out example to solidify our understanding. Suppose we have two objects with masses of 2 kg and 3 kg, respectively. Initially, the objects have velocities of 10 m/s and 5 m/s, respectively. After the collision, their velocities change to 6 m/s and 8 m/s, respectively. We want to find the energy lost during the collision.

First, let’s calculate the initial kinetic energy for each object:

Object 1:
$K.E. = \frac{1}{2} \times 2 \times 10^2 = 100 \text{ J}$

Object 2:
$K.E. = \frac{1}{2} \times 3 \times 5^2 = 37.5 \text{ J}$

Next, we can calculate the final kinetic energy for each object:

Object 1:
$K.E. = \frac{1}{2} \times 2 \times 6^2 = 36 \text{ J}$

Object 2:
$K.E. = \frac{1}{2} \times 3 \times 8^2 = 96 \text{ J}$

Now, let’s find the energy lost:

Object 1: $100 \text{ J} - 36 \text{ J} = 64 \text{ J}$

Object 2: $37.5 \text{ J} - 96 \text{ J} = -58.5 \text{ J}$

In this example, the energy lost in the collision is 64 J for object 1 and -58.5 J for object 2. The negative sign indicates that object 2 gained energy during the collision.

Special Cases in Energy Loss Calculation

Finding Kinetic Energy Lost in Elastic Collision

In elastic collisions, kinetic energy is conserved. This means that the total initial kinetic energy will be equal to the total final kinetic energy. Therefore, in an elastic collision, the energy lost is zero.

Calculating Mechanical Energy Lost in a Collision

In some collisions, such as those involving friction, mechanical energy is not conserved. Mechanical energy includes both kinetic energy and potential energy. To calculate the mechanical energy lost during a collision, we need to consider the changes in both kinetic and potential energy.

Estimating Thermal Energy Lost in Collision

Collisions involving deformations or interactions with a medium often result in the conversion of kinetic energy into thermal energy. To estimate the thermal energy lost in such collisions, additional factors like material properties and heat transfer mechanisms need to be considered. This estimation may involve complex calculations and empirical data.

Practical Applications of Energy Loss Calculation in Real Life

Role of Energy Loss Calculation in Vehicle Safety

Energy loss calculations play a crucial role in evaluating the safety of vehicle collisions. By understanding how much energy is dissipated during a crash, engineers can design safer vehicles and implement effective safety measures to mitigate the impact on occupants.

Energy Loss Calculation in Sports Science

In sports science, energy loss calculations are used to analyze and improve athletic performance. Understanding how much energy is lost during collisions, such as in contact sports or impacts between athletes, helps coaches and trainers develop strategies to minimize injuries and optimize performance.

Energy Loss Calculation in Environmental Physics

Energy loss calculations are also important in environmental physics. They help in studying the effects of collisions in natural phenomena, such as asteroid impacts, volcanic eruptions, and seismic activities. By quantifying the energy lost during these events, scientists can gain valuable insights into their consequences and make predictions for future scenarios.

Numerical Problems on how to find energy lost in a collision

Problem 1:

A car of mass 1000 kg collides with a stationary truck of mass 2000 kg. The car was initially moving with a velocity of 20 m/s, and after the collision, it moves with a velocity of 10 m/s. Find the energy lost during the collision.

Solution:

Given:
Mass of the car, m1 = 1000 kg
Mass of the truck, m2 = 2000 kg
Initial velocity of the car, u1 = 20 m/s
Final velocity of the car, v1 = 10 m/s

We can find the kinetic energy of the car before and after the collision using the formula:

$KE = \frac{1}{2} m v^2$

The initial kinetic energy of the car, KE1 = $\frac{1}{2} m1 u1^2$
The final kinetic energy of the car, KE2 = $\frac{1}{2} m1 v1^2$

The energy lost during the collision can be calculated as:

$Energy\,Lost = KE1 - KE2$

Substituting the given values, we have:

$Energy\,Lost = \frac{1}{2} m1 u1^2 - \frac{1}{2} m1 v1^2$

$Energy\,Lost = \frac{1}{2} \cdot 1000 \cdot (20^2 - 10^2)$

$Energy\,Lost = \frac{1}{2} \cdot 1000 \cdot (400 - 100)$

$Energy\,Lost = \frac{1}{2} \cdot 1000 \cdot 300$

$Energy\,Lost = 150000 \,J$

Therefore, the energy lost during the collision is 150,000 J.

Problem 2:

Two billiard balls, each having a mass of 0.2 kg, collide head-on. The initial velocity of the first ball is 5 m/s, and the initial velocity of the second ball is -3 m/s. After the collision, the first ball moves with a velocity of -2 m/s. Find the energy lost during the collision.

Solution:

Given:
Mass of each billiard ball, m = 0.2 kg
Initial velocity of the first ball, u1 = 5 m/s
Initial velocity of the second ball, u2 = -3 m/s
Final velocity of the first ball, v1 = -2 m/s

Similar to the previous problem, we can find the kinetic energy of each ball before and after the collision.

The initial kinetic energy of the first ball, KE1 = $\frac{1}{2} m u1^2$
The final kinetic energy of the first ball, KE2 = $\frac{1}{2} m v1^2$

The initial kinetic energy of the second ball, KE3 = $\frac{1}{2} m u2^2$
The final kinetic energy of the second ball, KE4 = $\frac{1}{2} m v2^2$

The energy lost during the collision can be calculated as:

$Energy\,Lost = (KE1 + KE3) - (KE2 + KE4)$

Substituting the given values, we have:

$Energy\,Lost = \left( \frac{1}{2} m u1^2 + \frac{1}{2} m u2^2 \right) - \left( \frac{1}{2} m v1^2 + \frac{1}{2} m v2^2 \right)$

$Energy\,Lost = \left( \frac{1}{2} \cdot 0.2 \cdot 5^2 + \frac{1}{2} \cdot 0.2 \cdot (-3)^2 \right) - \left( \frac{1}{2} \cdot 0.2 \cdot (-2)^2 + \frac{1}{2} \cdot 0.2 \cdot v2^2 \right)$

Simplifying further,

$Energy\,Lost = (0.5 + 0.18) - (0.2 + 0.02)$

$Energy\,Lost = 0.68 - 0.22$

$Energy\,Lost = 0.46 \,J$

Therefore, the energy lost during the collision is 0.46 J.

Problem 3:

A tennis ball of mass 0.1 kg collides with a wall. The ball was initially moving towards the wall with a velocity of 15 m/s, and after the collision, it bounces back with a velocity of -10 m/s. Find the energy lost during the collision.

Solution:

Given:
Mass of the tennis ball, m = 0.1 kg
Initial velocity of the ball, u = 15 m/s
Final velocity of the ball, v = -10 m/s

Using the same formula as before, we can find the kinetic energy of the ball before and after the collision.

The initial kinetic energy of the ball, KE1 = $\frac{1}{2} m u^2$
The final kinetic energy of the ball, KE2 = $\frac{1}{2} m v^2$

The energy lost during the collision can be calculated as:

$Energy\,Lost = KE1 - KE2$

Substituting the given values, we have:

$Energy\,Lost = \frac{1}{2} m u^2 - \frac{1}{2} m v^2$

$Energy\,Lost = \frac{1}{2} \cdot 0.1 \cdot 15^2 - \frac{1}{2} \cdot 0.1 \cdot (-10)^2$

$Energy\,Lost = \frac{1}{2} \cdot 0.1 \cdot 225 - \frac{1}{2} \cdot 0.1 \cdot 100$

$Energy\,Lost = 11.25 - 5$

$Energy\,Lost = 6.25 \,J$

Therefore, the energy lost during the collision is 6.25 J.

Also Read:

TechieScience Core SME

The TechieScience Core SME Team is a group of experienced subject matter experts from diverse scientific and technical fields including Physics, Chemistry, Technology,Electronics & Electrical Engineering, Automotive, Mechanical Engineering. Our team collaborates to create high-quality, well-researched articles on a wide range of science and technology topics for the TechieScience.com website.

All Our Senior SME are having more than 7 Years of experience in the respective fields . They are either Working Industry Professionals or assocaited With different Universities. Refer Our Authors Page to get to know About our Core SMEs.

techiescience.com