How to Show Energy Conservations by Calculating the Kinetic and Potential Energy

What is Energy?

Energy is the capacity of an object or a system to do work on surrounding objects or systems. For example, the moving car on the road has the energy.

But, how to know if the car has energy?  Well, if it hits another stationary car (light enough) then the stationery car will definitely move up to some distance.  And, from the definition of work you know that if the application point of the force moves to the direction of force then “the work is done”.

 

What is Mechanical Energy?

If the capacity of the object to do work is due to its velocity or change in position (with respect to surrounding objects) then the object is said to have mechanical energy in it. The mechanical energy is of two types: kinetic and potential. The mechanical energy equations for the two types of energy are discussed below:

The kinetic energy calculation formula is KE=0.5*m*v2

And the potential energy calculation formula (with respect to the earth) is PE=m*g*h

Where,

m – Mass of the object

v – Velocity of the object

h – Height of the object from the ground level

g – Acceleration due to gravity (typically 9.81 m/sec2)

 

 

In the previous car example, if the car is running over a plane road (i.e., slope of the road is zero) then the car has only kinetic energy. If the car is parked at the top of a hilly road (i.e., slope of the road is non-zero) then it has only potential energy. And if the car is coming down from the top of the hilly road then it has both the kinetic as well as the potential energy in it.

Conservation of Mechanical Energy

We will take the same previous example of the car to calculate the potential and the kinetic energy at different height of the hilly road and to see how the conservation of energy is followed. We will assume here that the car engine is off, it is in neutral gear and the brake is not applied.  Also, the hilly road has a slope angle of θ.

At hill-top:

Fig.1: Mechanical Energy conservation at hill-top

 

Potential energy calculation:         PE=M*g*H

Kinetic energy calculation:            KE=0.5*M*0=0

So, total mechanical energy at hill-top = M*g*H + 0 = M*g*H

 

At half hill-top:

 

Fig.2: Mechanical Energy conservation at half hill-top

 

Potential energy calculation:         PE=M*g*H/2

Kinetic energy calculation:            KE=0.5*M*V12

 

For calculating the velocity at half hill-top, you have to use the following equation:

V12 = U2 + 2*f*s………………………………Eqn.1

Where,

U = Initial velocity at hill-top = 0

V1 = Final velocity at half hill-top

f = the component of the gravitational acceleration toward the car’s movement=g*Sinθ.

s = Distance travelled by the car from the hill-top to the half hill-top

= H/(2*Sinθ)

 

So, from the Eqn.1,

V12 = g*H

So, total mechanical energy at the half hill-top = M*g*H/2 + 0.5*M*g*H

= M*g*H

 

 

At hill-bottom (just before stopping of the car):

 

Fig.3: Mechanical Energy conservation at hill-bottom

 

Potential energy calculation:         PE=M*g*0=0

Kinetic energy calculation:            KE=0.5*M*V22

For calculating the velocity at hill-bottom, you have to use the following equation:

V22 = U2 + 2*f*s………………………………Eqn.2

 

Where,

U = Initial velocity at hill-top = 0

V1 = Final velocity at half hill-top

f = the component of the gravitational acceleration toward the car’s movement=g*Sinθ.

s = Distance travelled by the car from the hill-top to the half hill-bottom

= H/(Sinθ)

 

So, from the Eqn.2,

V22 = 2*g*H

So, total mechanical energy at the half hill-top = 0 + 0.5*M*2*g*H

= M*g*H

 

 

So you have seen how to calculate mechanical energy for different positions of the car and how the sum of the potential and kinetic energy remain constant or the total mechanical energy is conserved.


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