Problem — Estimating the kinetic energy of a car

Problem

Estimate the kinetic energy of a small city car travelling at $50\;\mathrm{km/h}$. Compare it with the gravitational potential energy the same car would have if lifted to a height of $15\;\mathrm{m}$.

Solution

Estimated data. Mass of a small car: $m \approx 1000\;\mathrm{kg}$. Speed: $v = 50\;\mathrm{km/h} = \dfrac{50}{3{,}6}\;\mathrm{m/s} \approx 13{,}9\;\mathrm{m/s}$.

Kinetic energy. $E_k = \frac{1}{2}\,m\,v^2 = \frac{1}{2}\cdot 1000\;\mathrm{kg}\cdot (13{,}9\;\mathrm{m/s})^2 \approx 9{,}6\cdot 10^4\;\mathrm{J}$ $\ev{E_k \sim 10^5\;\mathrm{J}}$

Comparison with potential energy. Lifted by $h = 15\;\mathrm{m}$: $E_p = m\,g\,h = 1000\;\mathrm{kg}\cdot 9{,}8\;\mathrm{m/s^2}\cdot 15\;\mathrm{m} \approx 1{,}5\cdot 10^5\;\mathrm{J}$

The two values are of the same order of magnitude: travelling at $50\;\mathrm{km/h}$ stores roughly the same energy that would be needed to lift the car to the top of a building of about $5$ storeys. This is why braking from that speed dissipates as much energy (as heat) as a fall from $15\;\mathrm{m}$.

Links

Topics: Work and energy Concepts: Kinetic energy · Gravitational potential energy Skills: Fermi estimates