Kinetic Energy Notes

Kinetic Energy

This note covers the concept of kinetic energy, its equation, and its relationship with mass and velocity.

Objectives

  • Investigate examples of kinetic energy.

  • Calculate kinetic energy, mass, or velocity using the kinetic energy equation.

  • Predict the effect of mass and velocity changes on kinetic energy using proportional reasoning.

Assessment Questions

  1. What does each symbol mean in the equation E_k = \frac{1}{2} mv^2?

  2. Translate the equation E_k = \frac{1}{2} mv^2 into a sentence.

  3. How much kinetic energy does a 1.0 kg mass have when traveling at 30 m/s? At twice this speed?

  4. How fast would a 2.0 kg mass have to move to have a kinetic energy of 1,000 J?

  5. What is the mass of an object with a kinetic energy of 500 J and a speed of 25 m/s?

Physics Terms

  • Kinetic Energy

  • Mechanical Energy

Equations

The kinetic energy of a moving object is one-half of the product of its mass and the square of its velocity.

E_k = \frac{1}{2} mv^2

Kinetic Energy: Energy of Motion

  • Kinetic energy is the energy due to motion.

  • It depends on an object's mass and speed.

  • Example: A 1-liter water bottle (1 kg) moving at 1 m/s (2.4 mph) has a kinetic energy of 0.5 joules.

Exploring Kinetic Energy

  • Kinetic energy (E_k or KE) is the energy of motion. Any object with mass that is moving possesses kinetic energy.

  • When catching a ball, your hand applies a force over a distance to stop the ball. This force multiplied by the distance represents the transfer of the ball's kinetic energy to your hand through work.

  • A more massive or faster-moving ball has more kinetic energy and is harder to stop.

Relationship Between Kinetic Energy and Mass

  • Kinetic energy is calculated using the equation: E_k = \frac{1}{2} mv^2

  • E_k = kinetic energy (J)

  • m = mass (kg)

  • v = speed (m/s)

  • Kinetic energy is proportional to mass. If you double the mass, you double the kinetic energy (linear relationship).

  • Example:

    • A 2 kg ball moving at 4 m/s has a kinetic energy of: E_k = \frac{1}{2} (2 \text{ kg}) (4 \text{ m/s})^2 = 16 \text{ J}

    • A 1 kg ball moving at 4 m/s has a kinetic energy of: E_k = \frac{1}{2} (1 \text{ kg}) (4 \text{ m/s})^2 = 8 \text{ J}

Examples and Applications

  • A 1.0 kg object moving at 10 m/s has a kinetic energy of 50 joules.

  • A 10-gram (0.010 kg) marble traveling at 10,000 m/s (typical speed for space debris) has a kinetic energy of 500,000 J. This is equivalent to a 2,000 kg car going 50 mph.

  • A 2000 kg car with a kinetic energy of 500,000 joules has a speed of approximately 22.36 m/s, which is about 50 mph (1 m/s = 2.237 mph).

  • A battery containing 500 J of energy, with perfect conversion to kinetic energy, can launch a 1.0 kg object at approximately 31.6 m/s (about 70 mph).

Effect of Changing Mass and Velocity

  • Doubling the mass of an object doubles its kinetic energy.

  • Tripling the mass of an object triples its kinetic energy.

  • Doubling the velocity of an object quadruples its kinetic energy.

  • Tripling the velocity of an object increases its kinetic energy by a factor of nine.

E_k = \frac{1}{2} m v^2

Kinetic energy increases as the square of the speed.

3^2 = 9

  • If a cart with 10 joules of kinetic energy has its mass and velocity doubled, its new kinetic energy will be eight times as much: 80 joules.

Deriving the Formula

  • Hypothesis: The kinetic energy of an object equals the work done to change its velocity from zero to v.

  • Work is force times distance: W = Fd

    • F = ma

    • W = mad

  • Distance traveled at constant acceleration: d = \frac{1}{2} a t^2

  • W = ma (\frac{1}{2} a t^2) = \frac{1}{2} m a^2 t^2

  • Velocity reached at time t: v = at

  • W = \frac{1}{2} m (at)^2 = \frac{1}{2} m v^2

  • The kinetic energy of an object equals the work needed to change its velocity from zero to v.

Kinetic Energy from Work

  • Work done (Fd) results in a kinetic energy increase (\frac{1}{2} m v^2).

Typical Kinetic Energies

Object

Mass (kg)

Speed (m/s)

Speed (mph)

Kinetic Energy (J)

Baseball

0.15

9

20.1

6.1

Arrow

0.01

100

224

50

Bowling Ball

8

8

17.9

256

Sprinter

75

10

22.4

3,750

Car at 30 mph

1,500

13.4

30

135,000

Car at 60 mph

1,500

26.8

60

540,000

Car at 90 mph

1,500

40.2

90

1,214,000

Small Meteorite

1

20,000

44,740

200,000,000

Assessment Solutions

  1. What does each of the symbols mean in this equation: E_k = \frac{1}{2} mv^2 ?

    • E_k (or KE) = kinetic energy in joules

    • m = mass in kg

    • v = speed in m/s

  2. Translate the equation E_k = \frac{1}{2} mv^2 into a sentence with the same meaning.

    • The kinetic energy of an object is one half of the product of its mass multiplied by the square of its velocity.

  3. How much kinetic energy does a 1.0 kg mass have when traveling at a speed of 30 m/s?

    • E_k = \frac{1}{2} mv^2 = \frac{1}{2} (1 \text{ kg})(30 \text{ m/s})^2 = 450 \text{ J}

    • At twice the speed:
      E_k = 4 \times (450 \text{ J}) = 1800 \text{ J}

  4. How fast would a 2.00 kg mass have to be moving to have a kinetic energy of 1,000 J?

  5. What is the mass of an object that has a kinetic energy of 500 joules and a speed of 25 m/s?