Newton's second law of motion: F = ma

Push a light cart, then a heavy cart

Push an empty shopping cart and a full one with the same force. The empty one speeds up much more. Push the empty one harder, and it speeds up more still.

• For a fixed mass, acceleration grows in step with the force.
• For a fixed force, acceleration falls as the mass grows.

Newton's second law

$F$ is the net force on the object. One newton is the force that gives $1\ \text{kg}$ an acceleration of $1\ \text{m/s}^2$.

The car in the picture has $3600\ \text{N}$ forward and $600\ \text{N}$ back: a net force of $3000\ \text{N}$ on $1200\ \text{kg}$, so $a = 2.5\ \text{m/s}^2$.

You should find that the acceleration rises in a straight line with the force, and that doubling the mass halves it, so the car takes twice as long to reach $20\ \text{m/s}$.

Watch out. Use the net force, not just the engine's push. With the forces balanced, $F = 0$ and $a = 0$, even if the car is moving.

A worked example

A $1100\ \text{kg}$ car has an engine force of $4200\ \text{N}$ and a resistance of $900\ \text{N}$. Find its acceleration and how long it takes to reach $27\ \text{m/s}$ from rest.

1. Net force. $4200\ \text{N} - 900\ \text{N} = 3300\ \text{N}$.
2. Acceleration. $a = \dfrac{3300\ \text{N}}{1100\ \text{kg}} = 3\ \text{m/s}^2$.
3. Time. From Chapter 1, $t = \dfrac{v}{a} = \dfrac{27\ \text{m/s}}{3\ \text{m/s}^2} = 9\ \text{s}$.

Practise with new numbers

Try it in Python

The program works out the car.

Remember these

• $F = ma$, with $F$ the net force.
• $1\ \text{N} = 1\ \text{kg m/s}^2$.
• More force, more acceleration; more mass, less.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What net force gives a $5\ \text{kg}$ box an acceleration of $4\ \text{m/s}^2$?

2. The same force acts on a $2\ \text{kg}$ and a $6\ \text{kg}$ trolley. How do their accelerations compare?

3. A car moves at a steady $25\ \text{m/s}$. What is the net force on it?

4. A force of $150\ \text{N}$ gives a sledge an acceleration of $0.6\ \text{m/s}^2$. What is its mass?