Physics 9702/13 — May/June 2020

Background Concept

Kinetic energy is the energy an object has because it is moving. For an object of mass $m$ moving at speed $v$, its kinetic energy is

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

The key point for estimation questions is that $E_k$ depends on $v^2$, so the speed matters a lot: doubling $v$ makes $E_k$ four times bigger.

Understanding the Question

A man is running in a straight line. No exact mass or speed is given, so you are expected to use typical values for a person and for running speed, calculate an approximate kinetic energy, and then choose the closest order of magnitude from the options: $10\ \text{J}$, $100\ \text{J}$, $1000\ \text{J}$, or $10\ 000\ \text{J}$.

Approach

1. Choose a sensible estimate for a man's mass (e.g. $60$ to $80\ \text{kg}$).
2. Choose a sensible estimate for running speed (a few $\text{m s}^{-1}$; sprinting might be nearer $10\ \text{m s}^{-1}$, steady running around $4$ to $6\ \text{m s}^{-1}$).
3. Substitute into $E_k = \tfrac{1}{2}mv^2$ and compare the result to the given options by order of magnitude.

Step-by-Step Reasoning

Take typical values:

• mass $m \approx 70\ \text{kg}$
• speed $v \approx 5\ \text{m s}^{-1}$

Calculate kinetic energy:

$$E_k = \frac{1}{2}mv^2$$ $$E_k \approx \frac{1}{2}(70)(5^2)$$ $$5^2 = 25 \quad \Rightarrow \quad E_k \approx 35 \times 25 = 875\ \text{J}$$

$875\ \text{J}$ is closest to $1000\ \text{J} = 10^3\ \text{J}$, so the best option is C.

(Quick check: even if you chose $m=60\ \text{kg}$ and $v=4\ \text{m s}^{-1}$, you would get $E_k \approx 0.5\times 60\times 16 \approx 480\ \text{J}$, still of order $10^3\ \text{J}$ rather than $10^2\ \text{J}$.)

Key Takeaways

• Use $E_k = \tfrac{1}{2}mv^2$.
• For estimation, pick realistic values and aim for the correct power of ten.
• Speed is especially important because kinetic energy scales with $v^2$.

Common Mistakes

• Using $E_k = mv^2$ (missing the factor of $\tfrac{1}{2}$).
• Using an unrealistic speed (e.g. $50\ \text{m s}^{-1}$) which would give a wildly wrong energy.
• Confusing speed in $\text{m s}^{-1}$ with other units (though in an MCQ estimate, consistency still matters).

Things to Be Careful About

• Typical running speeds are a few $\text{m s}^{-1}$; even a fast sprinter is only around $10\ \text{m s}^{-1}$.
• Don’t overthink precision: the options differ by factors of $10$, so you only need the right order of magnitude.
• Remember that squaring the speed changes the scale quickly, so choose $v$ carefully.