0.21 Phy1190: energy -- kinetic and mechanical energy  (Page 3/8)

d = 0.5*a*t^2

where

• d represents the distance traveled
• a represents the constant acceleration
• t represents time

Substitution

Therefore, by substitution we can write

W = f*d = 0.5*f*a*t*t, or

W = 0.5*m*a*a*t*t

We're almost there

Recognizing that a = v/t where v represents velocity and t represents time, we can write

W = 0.5*m*(v/t)*(v/t)*t*t

Canceling terms in the numerator and denominator, we can write

W = 0.5*m*v*v = 0.5*m*v^2

And we're there

Since the kinetic energy possessed by the object equals the work that was done on the object to produce the current velocity (assuming no energy loss inthe interval), we can write

KE = 0.5*m*v^2

where

• KE represents the kinetic energy possessed by the object
• m represents the mass of the object
• v represents the velocity of the object

Once again, even though I assumed constant acceleration to derive this equation, the equation is true regardless of the nature of the acceleration thatcaused the object to possess its current velocity.

KE varies as the square of the velocity

This is an extremely important equation, because it shows that the kinetic energy possessed by an object increases as the square of the velocity of theobject. For example, if you are struck by a falling book with a velocity of 2 m/s, it will do four times as much work in damaging your body as a book with avelocity of only 1 m/s.

If the driver of a car increases the speed of the car from 55 mph to 70 mph, this represents about a 27-percent increase in speed but it also representsabout a 62-percent increase in kinetic energy.

Miscellaneous facts about kinetic energy

Like work, kinetic energy is a scalar quantity. It has magnitude but not direction.

Also like work and potential energy, the standard unit of kinetic energy is the joule. However, there are many other units that are used to express energyincluding:

• 1 foot-pound = 1.36 joules
• 1 British thermal unit (Btu) = 1055.1 joules
• 1 kilowatt-hour = 3600000 joules
• 1 calorie = 4.18 joules
• 1 electron volt = 1.6*10^(-19) joules
• 1 hartree = 4.4*10^(-18) joules
• 1 ton of TNT = 4.18 *10^9 joules

Some sample calculations involving kinetic energy are presented later .

Mechanical energy

We know what it means to do work on an object. When a force is applied to an object causing a displacement of the object (such as lifting a small child andplacing that child into a feeding chair), work has been done on the object.

Force is a two-way street

A force doesn't exist in isolation. From Newton's third law, we know that for each force that is applied to one object, an equal and opposite force is appliedto the other object. When you lift the child and place the child into a feeding chair, you supply the force necessary to overcome the force of gravity and lift the child.(You also do the work necessary to increase the child's gravitational potential energy.)

Work is also a two-way street

In all cases where a force is applied to an object (work is done on the object), some other object must apply the force and do the work. That force maybe applied by a person, a motor, a beast of burden, rushing water, wind, etc.