What is kinetic energy?

Suppose an object has mass m and is moving with a speed v, we define kinetic energy (K) as

K =
1 / 2
mv2

The unit for m is kg, the unit for the speed is m/s, and the unit for k is joules.

When an object is in motion, it is capable of doing work. Does it make sense to you that the object is capable of doing work? Take a look at the carts below.

Kinetic energy

The cart on the left  is pushing the cart on the right and thus doing work on it or applying a force over a certain distance.

Derivation of kinetic energy formula

To derive the formula, we will need a few concepts.

  •    We will need the work formula.
Therefore, the 3 equations we need are

F = ma      equation 1

v2 = v02 + 2ad     equation 2

W = Fd     equation 3

We will combine these 3 equations, do some math, and finally arrive at the formula.

We will start with W = Fd since what we are looking is the work a moving object can do.

W = F × d

Using equation 3, substitute ma for F

W = ma × d      equation 4

Now, use equation 2 to solve for a

v2 = v02 + 2ad

v02 + 2ad = v2

Let v0 = 0. Why? Since the object is usually at rest before starting moving, the initial speed will be zero then.

02 + 2ad = v2

0 + 2ad = v2

2ad = v2

Divide both sides by 2d

a =
v2 / 2d

Replace the value of a in equation 4

W = m
v2 / 2d
d

W = m
v2 / 2

K = m
v2 / 2

K =
mv2 / 2

Notice that the speed is squared. What happens if we double the speed?

Let v be the speed of a moving object. Let speed =  2v after the speed is doubled. 

K = m
(2v)2 / 2

K = m
4v2 / 2

K = 4
mv2 / 2

The 4 that you see means that the kinetic energy is quadrupled.

So if you go from 40 miles per hour to 80 miles per hour, it will take four times as much work to stop the vehicle.

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