Work, Energy, and Simple Machines · Lesson 5 of 13
Kinetic Energy
“Speed enters the formula once and somehow leaves wearing a square.”
• Explain kinetic energy as energy due to motion. • Derive the expression for kinetic energy from work and motion equations. • Calculate kinetic energy using mass and velocity in SI units. • Analyse how mass and speed affect kinetic energy. • Use the work-energy theorem to solve stopping and acceleration problems.
A slowly rolled cricket ball may barely disturb a wicket, while a fast delivery can send the bails flying. A loaded cart moving at the same speed as an empty cart is also harder to stop. Motion gives both objects energy, but the amount depends on more than the fact that they are moving. It depends on how much mass is moving and how fast it is moving.
Kinetic energy is the energy possessed by an object because of its motion.
An object chosen to be at rest has zero kinetic energy. If a force makes it move, the force does positive work and the object gains kinetic energy. If a force slows it down, that force does negative work and its kinetic energy decreases.
Deriving the Expression for Kinetic Energy
Consider an object of mass m moving initially with velocity u. A constant net force acts in the direction of motion, producing constant acceleration a. After displacement s, the velocity becomes v. We begin with a motion equation that connects these quantities.
Rearranging for displacement gives:
The work done by the constant net force is W = F × s. Newton’s second law gives F = ma. Substituting both the force and displacement expressions allows acceleration to cancel.
According to the work-energy theorem, this work is the change in the object’s kinetic energy. When the object starts from rest, u is zero, so the final kinetic energy is:
Dependence on Mass and Speed
For objects moving at the same speed, kinetic energy is directly proportional to mass. Doubling mass doubles kinetic energy. For an object of fixed mass, kinetic energy is proportional to the square of speed. Doubling speed makes the kinetic energy four times as large, while tripling speed makes it nine times as large. This squared dependence explains why a modest increase in vehicle speed greatly increases the energy that brakes must remove.
| Change | Resulting Kinetic Energy |
|---|---|
| Mass doubled at the same speed | Twice as large |
| Mass tripled at the same speed | Three times as large |
| Speed doubled for the same mass | Four times as large |
| Speed tripled for the same mass | Nine times as large |
Problem
A vehicle of mass m has speed v. Compare its kinetic energy at speed v and speed 2v.
- 1.Initial kinetic energy is K = one-half mv².
- 2.Replace v with 2v for the new kinetic energy.
- 3.Knew = one-half m(2v)².
- 4.Since (2v)² = 4v², Knew = 4 × one-half mv².
- 5.The kinetic energy becomes four times its original value.
Converting Velocity Before Calculation
Mass must be in kilograms and velocity in metres per second when kinetic energy is required in joules. To convert kilometres per hour into metres per second, multiply by 5/18. This follows because one kilometre is one thousand metres and one hour is three thousand six hundred seconds.
Problem
A cricket ball of mass 0.2 kg moves at 154.8 km h⁻¹. Calculate its kinetic energy.
- 1.Convert the velocity: 154.8 × 5/18 = 43 m s⁻¹.
- 2.Write the formula: K = one-half mv².
- 3.Substitute: K = one-half × 0.2 kg × (43 m s⁻¹)².
- 4.Square the velocity: 43² = 1849 m² s⁻².
- 5.Multiply: K = 0.1 × 1849 = 184.9 J.
- 6.Therefore, the ball has approximately 184.9 J of kinetic energy.
Using Negative Work to Find an Initial Velocity
Problem
A 15000 kg aircraft is stopped over 100 m by a constant backward force of 367500 N. Find its speed just before the wire caught it.
- 1.Let the initial speed be v. Initial kinetic energy = one-half × 15000 × v².
- 2.The final speed is zero, so final kinetic energy = 0 J.
- 3.The force is opposite to displacement, so the work is negative.
- 4.Work by the wire = 367500 × (-100) = -36750000 J.
- 5.Apply W = Kfinal - Kinitial.
- 6.-36750000 = 0 - one-half × 15000 × v².
- 7.Cancel the negative signs and solve: v² = (2 × 36750000) ÷ 15000 = 4900.
- 8.Therefore, v = 70 m s⁻¹, which is 252 km h⁻¹.
Problem
Objects A and B have masses m and 4m but equal kinetic energy. Find the ratio of their speeds.
- 1.Write equality of kinetic energies: one-half m vA² = one-half (4m) vB².
- 2.Cancel one-half and m from both sides.
- 3.vA² = 4vB².
- 4.Take the positive square root because speed is a magnitude.
- 5.vA = 2vB, so vA : vB = 2 : 1.
Kinetic Energy and Position
Kinetic energy is determined by mass and speed, not directly by where an object is located. If an object moves with constant speed, its kinetic energy stays constant even while its position changes. Position may change potential energy, however. A car travelling at constant speed up a hill keeps the same kinetic energy but gains gravitational potential energy because the engine continues doing work.
Problem
A car has 90000 J of kinetic energy at speed v. How much net work is required to reach speed 2v?
- 1.At twice the speed, kinetic energy becomes four times as large.
- 2.Final kinetic energy = 4 × 90000 J = 360000 J.
- 3.Net work equals the change in kinetic energy.
- 4.W = 360000 J - 90000 J = 270000 J.
- 5.The work required is three times the original kinetic energy because the final value is four times the original.
In K = one-half mv², the complete velocity value is squared. Convert velocity to metres per second first, then square it. Do not square only the unit or forget that doubling speed produces four times the energy.
Quiz
Which description best matches Kinetic Energy?
Which term matches this description: Kinetic energy is the energy possessed by an object because of its motion.
Which statement is a key takeaway from this lesson?
Which additional statement is also a key takeaway from this lesson?
Which further statement is also a key takeaway from this lesson?
Practice Problems
- Calculate the kinetic energy of a 5 kg object moving at 6 m s⁻¹.
- A 1200 kg car increases its speed from 10 m s⁻¹ to 20 m s⁻¹. Find the change in kinetic energy.
- Compare the kinetic energies of equal masses moving at speeds v and 3v.
- A 2 kg object with 100 J of kinetic energy is stopped. Find the work done on it.
Key Takeaways
• Kinetic energy is the energy of motion. • K = one-half mv². • Kinetic energy is proportional to mass and to the square of speed. • Positive net work increases kinetic energy; negative net work decreases it. • SI units must be used before numerical substitution.