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# Kinetic Energy and Work Done

## Introduction

Kinetic energy is the energy related to motion. For an object which is stationary, there is no kinetic energy. In this article, we are not talking about any energy change. For example, water is converted to heat energy and thus steam when heated. We can find many such energy changes in the environment and day-to-day life, but they are out of scope in this article. Here we discuss the kinetic energy which is associated with objects in motion. A moving object acquires kinetic energy due to the force applied to it, which can be expressed by an equation

$\mathrm{KE=\frac{1}{2}mv^2}$

But the work depends on the force applied to the object and the distance. Work done is the product of applied exertion and distance traversed. This article studies kinetic energy and work and the theorem connecting both together with some examples.

## What Is Work?

Work and work done are primarily used in the same denotation. However, in Physics we use Work Done since it is the product of force and distance. Work has to be done to get the distance covered. Right? When we say work alone, it is a scope definition of what needs to be done, what all material needs to be there, and so on. But the word work, when coming to the context of Physics, can be treated as work done.

## What Is Work Done?

When a force F is exerted on an object, it moves either linearly or at an angle to the linear path. If the object moves a distance d at an angle $\mathrm{\vartheta}$, then the work down W is given by the equation

$\mathrm{W=Fd\:Cos(\vartheta)}$

**Fig:1 work done by a body**

As you see in the figure above, the engine is performing work on tyres against friction. At the same time, tyres do work against road friction and gravity. The frictional force acts against the motion of the truck we know. There is no external force acting on the vehicle. So the constructive work done by the engine is acting on tyres to cancel the frictional force and gravitational force and to move forward at a constant velocity of V. Since the truck is moving at a steady pace, there is no change in kinetic energy $\mathrm{(\Delta\:V=0)}$. Hence, there is no work done as per the work-energy theorem. The potential head (mgh), as can be noted, is growing as the truck climbs the hill, thus preserving the conservation law.

## Types of Work Done

There are three types of work defined. They are Positive, Negative, and Zero work. Positive work is when the displacement of the object is in the same direction as the applied force. Negative work is when the displacement of the object is in the opposite direction as the external force applied. Zero work is when displacement is nil, even when an external force is applied to the object.

## Examples

### When do we say Work is Negative Work?

An example is when you try to move a cart up the hill, imagine that due to the weight of the load in the cart, you are slipping down. The cart is rolling down slowly amidst your thrust on it upwards. This is a negative work scenario.

What is an example of positive work? A person climbing a tree is a positive work as the displacement is in the same direction as the force exerted by the person upwards.

Lastly, let us see a zero work example. It is said that zero work is done when a person stands with a bag motionless. The weight of the bag is acting downwards due to gravity, but since displacement is not there, the work done is zero. Moving an object is positive work.

### What Is Kinetic Energy?

This is the energy gained by an object while in motion. The energy associated with a stationary object is potential energy. Kinetic energy is there wherever there is movement. Examples are flying a kite, flying aircraft, flying birds, racing cars, spinning windmills, moving clouds, walking, swimming, running, orbiting satellites, waterfall, and so on. The pendulum movement of a clock is an example of the interplay of kinetic energy and potential energy. When the pendulum is at the bottom center, the potential energy is zero and the kinetic energy is maximum. At both ends, the kinetic energy is zero and the potential energy is maximum (mgh).

All other energy like chemical, thermal, mechanical, electrical, etc, is a combination of potential energy and kinetic energy. The sum total of these energies is a constant at any point in time, which is the law of conservation of energy.

### Work-Energy Theorem

The theorem states that the work done on an object is the change in kinetic energy of it. The work done is measured in Joules and it is represented in the equation as

$\mathrm{W=\Delta K=\frac{1}{2}m(v^2-u^2)=m\times a \times d=F \times d}$

## Solved Examples

SN | Functions Used In This Section | Equation |
---|---|---|

1 | Kinetic Energy | $\mathrm{KE=\frac{1}{2}mv^2}$ |

2 | Work Done | $\mathrm{W=Fd}$ |

A truck weighing 1500kg tries to stop with a brake force of 20000 Newtons while moving at 18 km/hr speed. How far will the truck move before it stops completely?

$\mathrm{Starting\:KE =\frac{1}{2}\times 1500 \times (18 \times 60 \times 60/1000)^2=18750\:Joules}$

Final KE = 0 since final velocity is zero

Change in KE = 18750 - 0 = 18750 Joules

Work done is by the braking energy (force) which is 20000N and this is equal to a change in KE

$\mathrm{F\times\:d=W=18750}$

$\mathrm{20000\times\:d=18750}$

$\mathrm{d=18750/20000=0.9375 \:metres}$

An object weighing 50kg is moving at 5m/s speed. A force of 200N is applied for a distance of 30 meters, in the same direction of movement. What would be the final velocity of the object?

$\mathrm{Work\:done =F\:\times\:d = 200\:\times\:30 = 6000\:Joules}$

Applying work energy theorem,

$\mathrm{6000 = 1/2\:\times\:50\:\times\:{v_f}^2 - 1/2 \:\times\:50\:\times\:5^2 }$

$\mathrm{6000\:+\:625\:=\:25\:\times{v_f^{2}}}$

$\mathrm{{v_f}^2=265}$

$\mathrm{Final\:Velocity,\:v_f=\sqrt{265}=16.27 m/s}$

## Conclusion

The article provides an overall concept of kinetic energy and work particularly since these are connected to each other by the work-energy theorem. Different types of work and solved examples are dealt with extensively. The article brings forth the distinction between work, work done, power, energy, and such related terms to get a complete understanding. As a prelude to the solved examples section the equations used are listed for reference in a table. FAQ and Summary wind up the additional information and pointers.

## FAQs

**Q1. How are energy and work related? **

Ans. When the kinetic energy of an object is changed it must be equivalent to the net work done on the object. For example, in a straight-line collision, the two bodies travel towards each other and a collision occurs. The total work done is equivalent to the force of impact times the distance traveled during the impact.

i.e. Average impact force x distance traveled = change in kinetic energy.

**Q2. If a ball is kicked and moved 50 meters with a constant velocity, what is the work done by the kicker?**

Ans. Since velocity is constant, the change in kinetic energy becomes zero. So the work done is also zero as per the work-energy theorem. $\mathrm{\Delta W=\Delta K E}$

**Q3. What are the different types of kinetic energy?**

Ans. Light and Sound, Electrical and Thermal, Mechanical movement.

**Q4. When a spring is stretched from an unstretched state to twice its length, what energy does it hold?**

Ans. Resistive forces such as spring tension hold potential energy when it is stretched.

**Q5. What is the SI unit of force and work?**

Ans. Force is measured in Newtons. SI unit is $\mathrm{kg\:m\:s^{-2}}$. Work is measured in Joules. SI unit is Newton meter or $\mathrm{kg \:m^2\:s^{-2}}$.

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