What is Kinetic Energy?
For us to perfectly understand how to calculate kinetic energy, we will need to first define and and then derive its formula.
Definition of Kinetic Energy: The kinetic energy of a body can be defined as the energy of the body due to motion.
The symbol for kinetic energy is KE or K.E.
We measure kinetic energy in Joules (J).
Kinetic energy is a scalar quantity.
The formula for calculating kinetic is (1/2)mv2
Derivation of Kinetic Energy (K.E) Formula
Assuming an object of mass m in kilogram has an initial velocity of u in meters per second. The object attains a final velocity of v in meters per second through a distance s in meters. We can derive the kinetic energy formula by applying the third equation of motion, the Second newton’s law of motion, and the formula for work done.
Step 1
Let’s get to work, start by making acceleration a subject of the formula from the equation v2 = u2 + 2as
We have an equation that says v2 = u2 + 2as
When we take the initial velocity u to the left-hand side, the above equation becomes
v2 – u2 = 2as
We can now divide both sides by 2s to make a subject of the formula [ (v2 – u2)/2s= 2as/2s ] to obtain
a = (v2 – u2)/2s
Now, remember that second Newton’s law says the force, f = ma [where m = mass and a = acceleration]
Hence, plugin a = (v2 – u2)/2s into f = ma
Which implies that
f = m x [ (v2 – u2)/2s ]
Therefore, we can break the right-hand side into
f = (1/2s) x m x (v2 – u2)
Remember that the formula for work done is w = f x s [ where w = work done, f = force, and s = distance]
Step 2
Hence, we can substitute f in the equation w = f x s into f = (1/2s) x m x (v2 – u2) to obtain
K.E = (1/2s) x m x (v2 – u2) x s
and s will cancel each other to leave us with
K.E = (1/2) x m x (v2 – u2)
If the initial velocity is zero (0) we will end up with
K.E = (1/2) x m x (v2 – 0)
Which is
K.E = (1/2) mv2 or 0.5mv2
Units of Kinetic Energy (K.E)
The units of kinetic energy are as follows:
- Joules
- Kilojoules
- Calories
- We also use Foot-pound as a unit of kinetic energy
- Other units for kinetic energy are watt-hour and electron volt
Examples: Solved Problems on How to Calculate Kinetic Energy
Here are solved problems on how to calculate kinetic energy to help improve your understanding of the topic:
Example 1
A body of mass 5 kilograms is acted upon by a constant force of 10 newtons for 4 seconds. Calculate the kinetic energy gained by the body.
Solution
Data
mass, m = 5 kg
force, f = 10 N
Time, t = 4 s
K.E=?
The formula for calculating K.E is (1/2)mv2
But we don’t have v and we need to apply the above data to find it
When we read the question again, we will see that
Initial velocity, u = 0
acceleration, a =f/m [Because f = ma]
Thus, a = 10/5 = 2 ms-2
Now, plugin t= 4 s, u=0, and a=2 ms-2 into the equation v = u+at to obtain
v = 0 + 2 x 4 = 8 m/s
Hence,
K.E = (1/2) mv2 = 0.5 x 5 x 82 = 160 J
Therefore, the kinetic energy of the body is 160 Joules.
Example 2
An express train of mass 500 kilograms is moving with a velocity of 180 kilometers per hour. What is the kinetic energy of the train?
Solution
Data
Mass of the train, m = 500 kilograms
Velocity of the train, v = 180 km/h = (180 x1000m)/ 60 x 60s = 180,000m/3,600s = 50 m/s
And the formula for calculating kinetic energy is K.E = (1/2) mv2
Hence, we substitute our data into the above formula
K.E = 0.5 x 500 x 502 = 625,000 Joules = 625 kJ
Therefore, the kinetic energy of the train is 625 kilojoules
Example 3
A hydrogen molecule of mass 3.3 x 10-27 kilograms is moving with a velocity of 1.7 x103 m/s. Calculate its kinetic energy.
Solution
Data
Mass of the hydrogen, m = 3.3 x 10-27 kg
Velocity of hydrogen, v = 1.7 x 103 m/s
We can now apply the formula for K.E to solve the problem
K.E = (1/2) mv2
We can insert our data into the above formula to find the kinetic energy
K.E = 0.5 x 3.3 x 10-27 x (1.7 x 103)2
Which implies that K.E = 4.7685 x 10-21 Joules
Example 4
A stone of mass 5 kilograms is thrown vertically upward with a velocity of 10m/s. Find the kinetic energy on reaching the ground.
Solution
Data
Extract your data from the question, it will make it easy for you to solve the problem.
Mass of the stone, m = 5 kg
velocity, v = 10 m/s
K.E =?
The formula for calculating kinetic energy is K.E = 0.5 mv2
Thus,
K.E = 0.5 x 5 x 102 = 2.5 x 100 = 250 J
Therefore, the kinetic energy is 250 Joules
Example 5
A ball of mass 1000 grams is dropped from a height of 5 meters and rebounded to a height of 25 meters. Calculate the kinetic energy of the ball before impact.
Solution
Data
Mass of the ball, m = 1000 grams = 1 kg
height before impact = 5 meters
To find the velocity before impact, we need to understand that the ball is falling on the ground. Thus, we can apply the formula for potential energy
Potential energy (P.E) = mgh [ where m = mass, g = gravitational acceleration, and h = height]
We also know that K.E = P.E
Hence,
(1/2) mv2 = mgh
and m will cancel each other to give
(1/2) v2 = gh
But g is constant and is 9.8ms-2
Therefore,
v2 = 2gh = 2 x 9.8 x 5 = 98
Now, we can apply a square root to each side
v = √98 = 9.9 or 10 m/s which is the velocity before impact
Let’s insert our v above into the formula for kinetic energy
The kinetic energy is, K.E = 0.5 mv2 = 0.5 x 1 x 102 = 50 Joules
Example 6
A body of mass 20 kilograms, initially at rest is subjected to a force of 40 newtons for one second. Calculate the change in kinetic energy.
Solution
Data
The change in kinetic energy is the same as the work done on the body.
Hence
Mass = 20 kg
The force f = 40 Newtons
Time, t = 1 Second
K.E = (1/2)mv2
and v = u + at [where a = acceleration and u = initial velocity]
u = 0 because the body was at rest
a = f/m = 40/20 = 2ms-2
Hence
v = 0 + 2 x 1 = 2 m/s
Therefore,
K.E = 0.5 x 20 x 22 = 10 x 4 = 40 J
Therefore, the kinetic energy of the body is 40 Joules
Example 7
A body of mass 0.6 kg is thrown vertically upward from the ground with a speed of 20 m/s. Calculate its kinetic energy just before it hits the ground.
Solution
Data
Mass of the body, m = 0.6 kg
Velocity, v = 20 m/s
K.E =?
We can now apply the formula for kinetic energy which is K.E=(1/2)mv2 to solve the problem
After inserting our data into the above formula. We will get
K.E = 0.5 x 0.6 x 202 = 0.3 x 400 = 120 J
Therefore the kinetic energy before hitting the ground is 120 Joules
Example 8
A body of mass 5 kilograms falls from a height of 10 meters above the ground. What is the kinetic energy of the body just before it strikes the ground?
[Neglect energy losses and take g = 10 m/s2]
Solution
Data
Mass of the body, m = 5 kg
height is the distance, s = 10 m
K.E = (1/2)mv2
But we need to find v
Let us apply one of the equations of motion (v2 = u2 + 2as) to find v
After inserting our data into the above equation of motion, we will have
v2 = 02 + 2 x 9.8 x 10 [ Where a=g=10m/s2 and u=initial velocity= 0 because the body starts from rest]
v = √ 196 =14 m/s
Now we can apply the formula for kinetic energy
K.E = (1/2)mv2 = 0.5 x 5 x 142 = 490 Joules
Example 9
A bus of mass 800 kilograms is traveling at a speed of 5 m/s. Determine the kinetic energy of the bus.
Solution
Data
Mass of the bus, m = 800 kg
Speed of the bus, v = 5 m/s
K.E =?
Apply the formula below to solve the problem
K.E = (1/2)mv2
Substitute the equations above with our data
K.E = 0.5 x 800 x 52 = 400 x 25 = 10,000 J = 10 kJ
Therefore, the kinetic energy of the bus is 10 kilojoules
Example 10
A ball of mass 0.7 kilograms is thrown at a speed of 10 m/s. Calculate the kinetic energy of the ball
Solution
Data
Mass of the ball, m = 0.7 kg
Speed of the ball, v = 10 m/s
K.E =?
Solve the problem by applying K.E=(1/2)mv2
K.E = 0.5 x 0.7 x 102 = 0.35 x 100 = 35 J
Therefore, the kinetic energy of the ball is 35 Joules
Other Forms of Energy that Involves Kinetic Energy?
Kinetic energy can be:
- Mechanical
- Thermal
- In form of Sound
- Electrical or
- Radiation
Applications of Kinetic Energy
- To generate electricity
- Sailor working on the boat
- Firing a bullet
- Sending a rocket from one planet to another
Important Points on Kinetic Energy
- The energy is due to motion
- The energy is a horizontal energy
- Kinetic energy does not act against gravity
- The symbol of Kinetic energy is K.E or KE
You may also like to read:
How to Calculate Kinetic Energy Without Velocity
Mechanical Energy: Definition and Types
How to Calculate Work Done in Physics
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