Work and Kinetic Energy: Quick Revision

A brief note on Work and Kinetic Energy. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Scalar product: $\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$, a plain number.
• Work of a steady force: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$, in joules. Positive with the motion, zero across it, negative against it.
• Kinetic energy: $K = \tfrac12 mv^2 = \dfrac{p^2}{2m}$.
• Work-energy theorem: the net work equals the change in kinetic energy.
• A changing force: the work is the area under the force-position graph, $W = \int F\,dx$. A spring: $\tfrac12 k(x_2^2 - x_1^2)$.

The facts at a glance

• Joule: $1\ \text{J} = 1\ \text{N m} = 1\ \text{kg m}^2\,\text{s}^{-2}$
• Work from parts: $W = F_xd_x + F_yd_y + F_zd_z$
• Braking distance: $d = \dfrac{v^2}{2\mu g}$: twice the speed, four times the distance
• Spring from zero: $W = \tfrac12 kx^2$
• Momentum and $K$: $K = \dfrac{p^2}{2m}$

A worked example

A $1100\ \text{kg}$ car speeds up from rest to $16\ \text{m/s}$ over $120\ \text{m}$ of level road.

1. Kinetic energy gained. $\tfrac12 \times 1100\ \text{kg} \times (16\ \text{m/s})^2 = 140\,800\ \text{J}$.
2. Net work. By the theorem, $140\,800\ \text{J}$.
3. Average net force. $\dfrac{140\,800\ \text{J}}{120\ \text{m}} = 1173\ \text{N}$.

Practice with new numbers

A program to try

The program works out the work of a slanting pull and the speed it gives.

Mistakes to avoid

• Work with no displacement along the force. Carrying a bag level, or holding it still, does no work on it.
• Forgetting the sign. Friction and braking forces do negative work.
• $(v - u)^2$ for the change in $K$. It is $\tfrac12 m(v^2 - u^2)$: square first.
• $Fd$ for a changing force. Use the area under the graph.
• km/h in $\tfrac12 mv^2$. Change to m/s first.

Check yourself

1. A $20\ \text{N}$ pull at $30^\circ$ above the floor moves a toy $9.0\ \text{m}$. How much work does it do?

2. Find $\vec{A} \cdot \vec{B}$ for $\vec{A} = 2\hat{i} + 7\hat{j}$ and $\vec{B} = -3\hat{i} + \hat{j}$.

3. A $0.50\ \text{kg}$ ball's speed drops from $12\ \text{m/s}$ to $8.0\ \text{m/s}$. How much work did air resistance do?

4. A spring with $k = 600\ \text{N/m}$ is stretched from $0\ \text{cm}$ to $5.0\ \text{cm}$. How much work does it take?

5. Two cars have the same momentum. One has twice the mass of the other. Which has more kinetic energy?