Newton's second law — F equals ma — Physics

Before this

This leads to

Newton's third law
Newton's laws problems — pulleys and slopes

Same engine, different results

Put the same engine into a small car and into a loaded truck. Both engines push with the same force. Which one gets up to speed faster?

Everyone knows the answer: the small car. But how much faster? Twice as fast? Ten times? And what exactly do we mean by "faster to get up to speed"?

In the last lecture, the first law said that a resultant force changes an object's velocity, and that an object with more mass resists that change more strongly. It promised that this lecture would work out how large the change in velocity is for a given resultant force. Here is that answer. It is one equation, and it is probably the most used equation in all of mechanics.

Two experiments, one rule

Imagine a trolley on a smooth, level track, pulled by a steady force. You can measure its acceleration: how quickly its velocity changes, in metres per second, every second. Now change one thing at a time.

Experiment 1: keep the mass the same and change the force. Double the resultant force, and the acceleration doubles. Triple it, and the acceleration triples. Acceleration is proportional to the resultant force.
Experiment 2: keep the force the same and change the mass. Load the trolley with twice the mass, and the acceleration halves. Four times the mass gives a quarter of the acceleration. Acceleration is inversely proportional to the mass.

Acceleration against resultant force for three masses. Each line is straight and passes through the origin, so acceleration is proportional to force. The larger the mass, the less steep the line.

Put the two results together. The acceleration depends on the force divided by the mass: $a \propto F \div m$ . Rearranged, the force needed is proportional to mass times acceleration: $F \propto ma$ .

The last step is about units. Physicists chose the unit of force to make this proportion an equation. One newton (1 N) is defined as the resultant force that gives a mass of 1 kg an acceleration of 1 m/s². With that definition, the constant is exactly 1, and the rule becomes the famous equation.

Newton's second law of motion

The resultant force on an object equals its mass multiplied by its acceleration:

$F = ma$

$F$ is the resultant force in N, $m$ the mass in kg, and $a$ the acceleration in m/s². The acceleration is in the same direction as the resultant force.

Now the hook can be answered. If a loaded truck has 8 times the mass of the car, the same resultant force gives it one-eighth of the acceleration. It takes 8 times as long to reach the same speed. That is it. That is the whole idea. Force causes acceleration, and mass decides how much.

Three things to get right every time

1. $F$ is the resultant force. Never put a single force into $F = ma$ when other forces act too. Find the resultant first. A car with a driving force of 3600 N and resistive forces of 800 N has a resultant of 2800 N forwards. The 2800 N is what goes into the equation.

2. Direction matters. Force and acceleration both have a direction, and the acceleration always points the same way as the resultant force. Choose a positive direction at the start, and give every force a sign. Remember the skydiver from the last lecture: when her parachute opened, the resultant force pointed up, so her acceleration pointed up. She was still moving down, so she slowed down.

3. Mass is not weight. Mass is the amount of matter in an object, measured in kilograms. It is the same everywhere. Weight is the gravitational force on the object, measured in newtons:

W = m g

On Earth, $g = 9.8$ N/kg, so a 70 kg person has a weight of $70 \times 9.8 = 686$ N. On the Moon, $g$ is about 1.62 N/kg (NASA), so the same person weighs only about 113 N. Their mass is still 70 kg. Bathroom scales actually measure the force you press down with, and then divide by 9.8 to show kilograms. In everyday speech, "I weigh 70 kg" means "my mass is 70 kg".

Notice that $g$ has two units: N/kg and m/s². They are the same thing, because 1 N is 1 kg m/s². The second unit tells you something important, which the next box explores.

Stop and think

Drop a 5 kg ball and a 1 kg ball from the same height, with air resistance too small to matter. The 5 kg ball has five times the weight. Does it fall five times faster?

Show the answer

No. They land together. The 5 kg ball has five times the force on it, but it also has five times the mass to accelerate. Using $F = ma$ with the weight as the only force: $a = W \div m = m g \div m = g$ . The mass cancels. Every falling object, heavy or light, accelerates at $g = 9.8$ m/s² when air resistance can be ignored. That is why $g$ can be written in m/s².

Explore the equation below with a car of your own.

Explore — force, mass and acceleration

A car is moving at 10 m/s. Resistive forces on it are 800 N. Change the driving force, then change the mass. Watch the resultant force, the acceleration, and the speed 3 seconds later.

Driving force: 3600 N

Mass: 1200 kg

The car moves at 10 m/s with 800 N of resistive forces. The table shows four settings.

Driving force, mass	Acceleration	Speed after 3 s
3600 N, 1200 kg	≈ 2.33 m/s²	17.0 m/s
2000 N, 2000 kg	0.6 m/s²	11.8 m/s
800 N, 1200 kg	0	10.0 m/s
0 N, 800 kg	−1.0 m/s²	7.0 m/s

When the driving force equals the resistive forces, the resultant is zero and the speed does not change: the first law. Otherwise, the acceleration is the resultant (driving force − 800 N) divided by the mass.

Did you try setting the driving force to 800 N? The resultant is zero, the acceleration is zero, and the car keeps its speed. The first law is simply the second law with $F = 0$ . And did you notice what doubling the mass did? With the same resultant force, the acceleration halved, and the speed after 3 seconds grew by only half as much.

Worked examples

Worked Example 1 — A car speeding up

A 1200 kg car has a driving force of 3600 N and resistive forces of 800 N. Find its acceleration.

Take forwards as positive. Resultant force: $F = 3600 - 800 = 2800$ N.

$a = F \div m = 2800 \div 1200 \approx 2.33$ m/s², forwards.

Check: a positive resultant gives a positive acceleration, so the car speeds up. That matches a driving force larger than the resistance.

Worked Example 2 — A rocket at launch

A rocket of mass 50 000 kg must accelerate upwards at 15 m/s². What thrust must its engines produce?

The two forces on the rocket. Up is taken as positive. The arrows are not to scale.

Take up as positive. Weight: $W = m g = 50000 \times 9.8 = 490000$ N, downwards.

The resultant force is the thrust minus the weight, and it must equal $ma$ :

T - 490000 = 50000 \times 15 = 750000

So $T = 1240000$ N, or $1.24 \times 1 0^{6}$ N.

This is the step people rush. If you write $T = ma = 750000$ N, you have forgotten the weight. The engines must first hold the rocket up, and only the force left over accelerates it. Check: the thrust is larger than the weight, as it must be for a rocket that rises.

Worked Example 3 — Braking

A 1500 kg car travelling at 20 m/s brakes steadily and stops in 4.0 s. Find the braking force.

First the acceleration, from SUVAT with forwards positive: $v = u + a t$ , so $0 = 20 + a \times 4.0$ , and $a = - 5.0$ m/s².

Then the force: $F = ma = 1500 \times (- 5.0) = - 7500$ N.

The minus sign means the resultant force points backwards: 7500 N against the motion. Check: a slowing car must have a backward resultant force, and it does.

Worked Example 4 — On the Moon

An astronaut has a mass of 70 kg. Find her weight on Earth and on the Moon. Then find the resultant force needed to give her an acceleration of 2 m/s² sideways in each place.

Weight on Earth: $70 \times 9.8 = 686$ N. On the Moon: $70 \times 1.62 \approx 113$ N.

Sideways: $F = ma = 70 \times 2 = 140$ N in both places.

Her weight changes, but her mass, and so her inertia, does not. On the Moon she is easy to lift, but just as hard to push sideways into motion, or to stop.

Here is the rocket mistake in a different setting. Find it.

Spot the error — the tension in a lift cable

An 800 kg lift accelerates upwards at 1.5 m/s². A student finds the tension in its cable. Exactly one line is wrong. Tap it.

A student's working for the tension in the cable of an 800 kg lift accelerating upwards at 1.5 m/s². Line 4 is wrong.

Line 1. Take up as positive. Correct.
Line 2. The forces are the tension T upwards and the weight, 800 × 9.8 = 7840 N, downwards. Correct.
Line 3. The resultant force is ma = 800 × 1.5 = 1200 N, upwards. Correct.
Line 4. "So the tension is T = 1200 N." Wrong: the resultant is T − 7840, so T − 7840 = 1200 and T = 9040 N.
Line 5. Check: the tension must be larger than the weight, because the lift speeds up upwards. Correct, and this check shows that line 4 cannot be right.

Three classic mistakes

Trap 1 — One force instead of the resultant

$F = ma$ uses the resultant force. Using only the driving force, the thrust or the tension gives the wrong acceleration. List every force, give each a sign, and add them first.

Trap 2 — Weight in kilograms

Weight is a force, so it is measured in newtons: $W = m g$ . "A weight of 60 kg" mixes up the two quantities. Write "a mass of 60 kg, so a weight of 588 N".

Trap 3 — "More force means more speed"

A resultant force sets the acceleration, not the speed. A car can move fast with zero resultant force (steady speed), and a car that has just started can have a large resultant force at low speed. Force tells you how quickly the velocity changes.

Try it yourself

Take $g = 9.8$ N/kg. Work each one on paper first. Open a hint only when you are stuck.

A bat hits a 0.16 kg ball. During the hit, the resultant force on the ball is 300 N. Find the ball's acceleration.

Hint 1
Rearrange $F = ma$ to make $a$ the subject.

Hint 2
$a = 300 \div 0.16$ .

Full answer
$a = F \div m = 300 \div 0.16 = 1875$ m/s², in the direction of the force. That is huge, but it lasts only a tiny fraction of a second. Check: a small mass and a large force should give a very large acceleration.
A resultant force of 450 N gives a loaded cart an acceleration of 1.5 m/s². What is the mass of the cart?

Hint 1
Rearrange $F = ma$ to make $m$ the subject.

Hint 2
$m = 450 \div 1.5$ .

Full answer
$m = F \div a = 450 \div 1.5 = 300$ kg. Check: $300 \times 1.5 = 450$ N.
A 600 kg lift accelerates downwards at 2.0 m/s². Find the tension in its cable.

Hint 1
Take down as positive, since the acceleration is downwards. The weight is then positive and the tension negative.

Hint 2
$600 \times 9.8 - T = 600 \times 2.0$ .

Full answer
Weight $= 600 \times 9.8 = 5880$ N. With down as positive: $5880 - T = 600 \times 2.0 = 1200$ , so $T = 4680$ N. Check: the tension is less than the weight, which is what lets the lift speed up downwards.
A 70 kg skydiver is falling. (a) When the air resistance on her is 400 N, find her acceleration. (b) What is her acceleration when the air resistance reaches 686 N? (c) Just after her parachute opens, the air resistance is 1500 N. Find her acceleration, and describe her motion.

Hint 1
Her weight is $70 \times 9.8 = 686$ N. Take down as positive.

Hint 2
Resultant = 686 − air resistance; then divide by 70.

Full answer
(a) Resultant $= 686 - 400 = 286$ N down, so $a = 286 \div 70 \approx 4.1$ m/s² downwards: she is speeding up. (b) Resultant $= 686 - 686 = 0$ , so $a = 0$ : she is at terminal velocity, as in the first-law lecture. (c) Resultant $= 686 - 1500 = - 814$ N, so $a = - 814 \div 70 \approx - 11.6$ m/s². The minus sign means the acceleration is upwards. She is still moving down, so she is slowing down, quickly.

The bigger idea

$F = ma$ says three things at once. Forces cause accelerations, not motion itself. The acceleration points the same way as the resultant force. And mass is a measurable property of matter: its resistance to being accelerated. The same idea predicts a car's acceleration, the thrust a rocket needs and the path of a planet.

There is something strange hidden in the Stop-and-think. The mass that resists acceleration and the mass that gravity pulls on turn out to be exactly equal, which is why every object falls with the same acceleration. Newton could not explain why. Einstein, more than two hundred years later, built his theory of gravity on that fact. For everyday speeds, though, $F = ma$ works so well that engineers use it every day. The next lecture asks a question this one has avoided: when one object pushes another, what does the second object do back?

Key concepts in this lesson

Newton's second law says that the resultant force on an object equals its mass multiplied by its acceleration, F = ma, with the acceleration in the direction of the resultant force.
One newton is the resultant force that gives a mass of 1 kg an acceleration of 1 m/s².
The F in F = ma is always the resultant force, so every force must be included with its sign before using the equation.
Weight is the gravitational force on an object, W = mg, measured in newtons, while mass is measured in kilograms and is the same everywhere.
When weight is the only force, every object falls with the same acceleration g, because the mass cancels.