Kinetic Energy: Formula & Calculations
Understanding the energy of motion - its formula, derivation, and practical applications
What is Kinetic Energy?
Kinetic energy is the energy possessed by an object due to its motion. Any object that is moving has kinetic energy. The amount of kinetic energy depends on two factors:
Mass (m)
The heavier the object, the more kinetic energy it has when moving at the same speed.
Velocity (v)
The faster the object moves, the more kinetic energy it has. Note: KE depends on velocity squared!
The Kinetic Energy Formula
Kinetic Energy Formula
Where kinetic energy (KE) equals one-half times the mass (m) times the square of the velocity (v)
Derivation of the Formula
The kinetic energy formula can be derived from the work-energy theorem, which states that the work done on an object equals its change in kinetic energy.
Derivation Steps
Start with Work Done
Work done (W) on an object equals force (F) times displacement (s) in the direction of the force:
Apply Newton's Second Law
Force equals mass times acceleration (F = m × a). Substitute into the work equation:
Use Kinematics Equation
From kinematics, for constant acceleration starting from rest: v² = u² + 2as, where u=0, so v² = 2as, therefore a = v²/(2s):
Simplify the Equation
The displacement (s) cancels out:
Work Equals Change in Kinetic Energy
By the work-energy theorem, work done equals change in kinetic energy. Starting from rest, this gives us the kinetic energy formula:
Graphical Representation
The relationship between kinetic energy and velocity is quadratic, while the relationship with mass is linear.
Kinetic Energy vs. Velocity
KE increases with the square of velocity. Doubling velocity quadruples kinetic energy!
Kinetic Energy vs. Mass
KE increases linearly with mass. Doubling mass doubles kinetic energy (at constant velocity).
Interactive Kinetic Energy Demo
Adjust the mass and velocity sliders to see how they affect kinetic energy in real-time.
Calculated Kinetic Energy
KE = ½ × 10 kg × (10 m/s)²
Example Problems
Example 1: Calculating Kinetic Energy
A car with a mass of 800 kg is traveling at a speed of 25 m/s. Calculate its kinetic energy.
Step 1: Write the formula
KE = ½ × m × v²
Step 2: Substitute known values
m = 800 kg, v = 25 m/s
KE = ½ × 800 × (25)²
Step 3: Calculate velocity squared
(25)² = 25 × 25 = 625
KE = ½ × 800 × 625
Step 4: Perform multiplication
½ × 800 = 400
400 × 625 = 250,000
Step 5: State the answer
KE = 250,000 J or 250 kJ
Example 2: Finding Mass from Kinetic Energy
A car has a kinetic energy store of 64,800 J. It is travelling at a speed of 12 m/s. Calculate its mass.
Step 1: Write the formula
KE = ½ × m × v²
Step 2: Rearrange to solve for mass (m)
m = (2 × KE) / v²
Step 3: Substitute known values
KE = 64,800 J, v = 12 m/s
m = (2 × 64,800) / (12)²
Step 4: Calculate denominator
(12)² = 12 × 12 = 144
m = (129,600) / 144
Step 5: Perform division
129,600 ÷ 144 = 900
Step 6: State the answer
Mass = 900 kg
Example 3: Finding Velocity from Kinetic Energy
A 0.5 kg ball has 100 J of kinetic energy. Calculate its velocity.
Step 1: Write the formula
KE = ½ × m × v²
Step 2: Rearrange to solve for velocity (v)
v² = (2 × KE) / m
v = √[(2 × KE) / m]
Step 3: Substitute known values
KE = 100 J, m = 0.5 kg
v = √[(2 × 100) / 0.5]
Step 4: Calculate numerator
2 × 100 = 200
v = √[200 / 0.5]
Step 5: Perform division
200 ÷ 0.5 = 400
v = √400
Step 6: Calculate square root
√400 = 20
Step 7: State the answer
Velocity = 20 m/s
Kinetic Energy Calculator
Use this calculator to solve for any variable in the kinetic energy equation.