Formula Grids · Mechanics · M-15

Ek = ½mv² Kinetic energy

Calculation grid60 cellsTarget: 10 min

\(E_k = \tfrac12 mv^2\)

\(E_k\): kinetic energy in J  ·  \(m\): mass in kg  ·  \(v\): speed in m/s

  • Square \(v\) first, then multiply by \(m\), then halve. Energy is a scalar: the direction does not matter.
  • Units are in the column and row headings, so write numbers only in the cells.

A Find the kinetic energy \(E_k\) \(E_k = \tfrac12 mv^2\)  (answers in J)

mass ↓  velocity → 1 m/s2 m/s3 m/s4 m/s−2 m/s
1 kg
2 kg
4 kg
10 kg

Notice: Across a row, \(v\) doubles (1 → 2 m/s), so \(E_k\) becomes times larger. Why is the last column positive?

B Find the speed \(v\) \(v = \sqrt{2E_k \div m}\)  (answers in m/s)

mass ↓  \(E_k\) → 4 J16 J36 J100 J1.6 kJ
0.5 kg
2 kg
8 kg
50 kg

Notice: Down a column, the mass is ×4 (0.5 → 2 → 8 kg) and the speed is ÷.

C Find the mass \(m\) \(m = 2E_k \div v^2\)  (answers in kg)

velocity ↓  \(E_k\) → 10 J50 J100 J200 J1 kJ
1 m/s
2 m/s
5 m/s
10 m/s

Notice: From 1 m/s to 2 m/s, the mass for the same energy is ÷. Speed matters more than mass.

Think about it Concept check · circle one answer

Concept: how \(E_k\) depends on \(v\)

1. The speed of a car doubles. Its kinetic energy becomes

  1. 2 times
  2. 4 times
  3. 8 times
  4. the same

\(E_k \propto v^2\): \(2^2 = 4\). This is why speed matters so much in crashes.

Concept: energy is a scalar

2. Two identical cars move at 10 m/s, one north and one south. Their kinetic energies are

  1. equal
  2. opposite
  3. one is negative
  4. zero in total

\(v^2\) is always positive, and energy has no direction. Both cars have the same \(E_k\).

Concept: the unit of energy

3. Using \(E_k = \tfrac12 mv^2\), the joule is equal to

  1. kg·m/s
  2. kg·m²/s²
  3. kg·m/s²
  4. N/m

kg × (m/s)² = kg·m²/s² = (kg·m/s²)·m = N·m = J.

Formula Grids · M-15 Ek = ½mv² · Set 1Calculator: not needed