Chapter 4 of 8

All 8 chapters
  1. 1Scalars and vectors, readunderstood
  2. 2Velocity and acceleration, readunderstood
  3. 3Newton's first law, readunderstood
  4. 4Newton's second law, readunderstood
  5. 5Weight and falling, readunderstood
  6. 6Newton's third law, readunderstood
  7. 7Momentum, readunderstood
  8. 8Stopping distances, readunderstood

S4.8.4 Chapter 4

Newton's second law

An empty truck picks up speed quickly. A loaded truck with the same engine picks up speed slowly. Why?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use force equals mass times acceleration in calculations
  • Explain why a loaded truck speeds up more slowly than an empty one

The first law says that with no resultant force, the velocity stays the same. What happens when there is a resultant force? The velocity changes: the object accelerates. The second law says by how much.

Two things decide the acceleration. One is the resultant force. A bigger force on the same object gives a bigger acceleration. The other is the mass. The same force on a bigger mass gives a smaller acceleration.

Newton’s second law: the resultant force on an object equals its mass times its acceleration. In symbols, F = m × a. F is in newtons, N. m is in kilograms, kg. a is in metres per second squared, m/s².

The acceleration points in the same direction as the resultant force. A force to the right makes an object speed up if it moves right, and slow down if it moves left.

The newton is defined by this law. One newton is the resultant force that gives a mass of 1 kg an acceleration of 1 m/s². So 1 N = 1 kg m/s².

Take a trolley of mass 2 kg. We pull it with different resultant forces and measure its acceleration. Plot the results.

Draw it first

Acceleration against force for a 2 kg trolley

  1. A graph with the axes acceleration (m/s²) and force (N). Force runs from 0 to 10 N and acceleration from 0 to 5 m/s². Step 1: dots at 2 N, 1 m/s² and 4 N, 2 m/s². Step 2: dots at 6 N, 3 m/s², 8 N, 4 m/s² and 10 N, 5 m/s². Step 3: a straight line from the start through every dot.246810123450force (N)acceleration (m/s²)

    1 A force of 2 N gives an acceleration of 1 m/s². A force of 4 N gives 2 m/s². Mark both dots.

  2. A graph with the axes acceleration (m/s²) and force (N). Force runs from 0 to 10 N and acceleration from 0 to 5 m/s². Step 1: dots at 2 N, 1 m/s² and 4 N, 2 m/s². Step 2: dots at 6 N, 3 m/s², 8 N, 4 m/s² and 10 N, 5 m/s². Step 3: a straight line from the start through every dot.246810123450force (N)acceleration (m/s²)

    2 Forces of 6 N, 8 N and 10 N give accelerations of 3, 4 and 5 m/s². Mark each dot.

  3. A graph with the axes acceleration (m/s²) and force (N). Force runs from 0 to 10 N and acceleration from 0 to 5 m/s². Step 1: dots at 2 N, 1 m/s² and 4 N, 2 m/s². Step 2: dots at 6 N, 3 m/s², 8 N, 4 m/s² and 10 N, 5 m/s². Step 3: a straight line from the start through every dot.246810123450force (N)acceleration (m/s²)

    3 The dots lie on a straight line through the origin. Double the force and the acceleration doubles.

Now keep the force fixed at 12 N and change the mass. The more mass, the smaller the acceleration. Double the mass and the acceleration halves.

A bar chart called acceleration (m/s²) for a resultant force of 12 N on different masses. The bar for 1 kg reaches 12. The bar for 2 kg reaches 6. The bar for 3 kg reaches 4. The bar for 4 kg reaches 3. The bar for 6 kg reaches 2. The scale goes from 0 to 12 in steps of 2.0246810121 kg2 kg3 kg4 kg6 kgacceleration (m/s²)
The same force of 12 N on a bigger mass gives a smaller acceleration.

We can rearrange F = m × a. To find the acceleration, divide the force by the mass: a = F ÷ m. To find the mass, divide the force by the acceleration: m = F ÷ a.

See it three ways

As a picture

A block labelled 4 kg with one arrow to the right labelled resultant, 20 N. A bar marked bar = 10 N shows the scale.4 kgresultant20 Nbar = 10 N

In words

The resultant force on an object is its mass times its acceleration. The same force gives a bigger mass a smaller acceleration.

In symbols

a = F ÷ m = 20 N ÷ 4 kg = 5 m/s²

Always use the resultant force. If the engine pulls forward and drag pulls back, take one from the other first. Always use the mass in kilograms. A mass in grams must be divided by 1 000 first.

Worked example

A small car has a mass of 800 kg. Its engine gives a forward force of 2 400 N. Resistance pulls back with 800 N. Find its acceleration.

  1. The resultant force is 2 400 N − 800 N = 1 600 N, forward.
  2. Use a = F ÷ m, with F = 1 600 N and m = 800 kg.
  3. a = 1 600 N ÷ 800 kg = 2 m/s², forward.

Answer: 2 m/s²

A car with two arrows. An arrow to the right is labelled engine, 2 400 N. A shorter arrow to the left is labelled resistance, 800 N. A bar marked bar = 800 N shows the scale.engine2 400 Nresistance800 Nbar = 800 N
The arrows are to scale. The resultant is the long arrow minus the short one.

Worked example

A sprinter has a mass of 60 kg. She accelerates at 5 m/s² at the start of a race. What resultant force acts on her?

  1. Use F = m × a, with m = 60 kg and a = 5 m/s².
  2. F = 60 kg × 5 m/s² = 300 kg m/s².
  3. One kg m/s² is one newton, so F = 300 N.

Answer: 300 N

A block labelled sprinter with one arrow to the right labelled resultant, 300 N. A bar marked bar = 100 N shows the scale.sprinterresultant300 Nbar = 100 N
The resultant force on the sprinter points the way she accelerates.

Now the trucks. An empty truck has a mass of 4 000 kg. Loaded, it has a mass of 12 000 kg. In both cases the resultant force is 6 000 N.

Worked example

A truck has a mass of 4 000 kg empty and 12 000 kg loaded. The resultant force is 6 000 N in both cases. Find each acceleration, and say how the two compare.

  1. The resultant force is the same for both: 6 000 N.
  2. Empty truck: a = 6 000 N ÷ 4 000 kg = 1.5 m/s².
  3. Loaded truck: a = 6 000 N ÷ 12 000 kg = 0.5 m/s².
  4. The loaded mass is 12 000 ÷ 4 000 = 3 times bigger, so the acceleration is 3 times smaller.

Answer: Empty 1.5 m/s², loaded 0.5 m/s²

A bar chart called acceleration (m/s²). The bar for empty reaches 1.5. The bar for loaded reaches 0.5. The scale goes from 0 to 2 in steps of 0.5.00.511.52emptyloadedacceleration (m/s²)
The same resultant force of 6 000 N. The loaded truck has three times the mass and a third of the acceleration.

That answers the opening question. The same engine can give the same force whatever the load. A loaded truck has more mass, so by a = F ÷ m it speeds up more slowly. It also needs a bigger force than an empty truck to slow down at the same rate.

Try it

The same pull on a light box and a heavy box

You need: a small box with a string loop, a rubber band, stones to load the box, a smooth, level floor, a friend to count slow beats.

Be safe: Hold the band low and away from your face. A band that snaps or slips off can sting. Keep the floor clear so that nobody trips. Keep the stones away from small children.

  1. Tie the rubber band to the string loop. Mark two dots on the band, 3 cm apart. When you pull, keep the dots 6 cm apart. This is your stretch.
  2. Lay the empty box on the floor. Pull the band so that the two dots stay 6 cm apart. A friend counts two slow beats while you walk.
  3. Mark where the box is after two beats. Measure how far it went from the start.
  4. Now put the stones in the box. Pull with the same stretch for two beats and measure again.

What you will see: The same stretch gives about the same pull. The loaded box goes much less far in the same time. It has more mass, so the same force gives it a smaller acceleration. The stones also add friction, which makes the difference bigger.

If you cannot try it, this is what happens.
A table of one set of results, with the columns Box and Distance. The empty box: 90 cm. The loaded box: 25 cm.BoxDistanceempty90 cmloaded25 cm

A wrong idea many people have

A bigger force always gives a bigger velocity.

Why it feels right: A harder push on a swing or a cart seems to make it go faster at once.

The test that breaks it: A car cruises at 20 m/s. The resultant force on it is 0 N, because the drive equals the drag. By F = m × a the acceleration is 0, yet the velocity is large.

What is true: The resultant force sets the acceleration, not the velocity. A steady resultant force makes the velocity grow steadily, for as long as it acts.

A wrong idea many people have

In F = m × a, the force is the push of the engine.

Why it feels right: The engine is the force we think about, and it is the biggest force in the picture.

The test that breaks it: The small car has an engine force of 2 400 N and resistance of 800 N. Using 2 400 N gives a = 3 m/s². The real acceleration is 2 m/s². Only the resultant, 1 600 N, gives that.

What is true: F is always the resultant force on the object. Add the forces first, with their directions. Then put the resultant into F = m × a.

A wrong idea many people have

You can put a mass in grams straight into F = m × a.

Why it feels right: Grams and kilograms are both units of mass, and the sum still looks correct.

The test that breaks it: A ball of 500 g has a resultant force of 2 N. Using 500 gives a = 2 ÷ 500 = 0.004. In kilograms, 500 g is 0.5 kg, and a = 2 ÷ 0.5 = 4 m/s².

What is true: The newton is defined with the kilogram. Change grams to kilograms first, by dividing by 1 000.

Key words

second law
the resultant force on an object equals its mass times its acceleration: F = m × a.
newton
the unit of force. One newton gives a mass of 1 kg an acceleration of 1 m/s².
resultant force
the one force that has the same effect as all the forces on an object together.

Back to your guess

The answer is A: The same force gives a bigger mass a smaller acceleration.

The engine gives the same force in both cases. The loaded truck has more mass, so the same resultant force gives it a smaller acceleration. This is Newton’s second law. A loaded truck does have a little more rolling friction, but the main reason is its mass.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A resultant force of 36 N acts on a trolley of mass 4 kg. What is its acceleration?

    Tap your answer.

    1. Yes, that is right. a = F ÷ m = 36 N ÷ 4 kg = 9 m/s².

    2. Not this one. You multiplied the force by the mass. To find the acceleration, divide the force by the mass. Try another.

    3. Not this one. You divided the mass by the force. It is the force divided by the mass. Try another.

    4. Not this one. You added the force and the mass. The second law is a division, a = F ÷ m. Try another.

    See the answer

    9 m/s²

    a = F ÷ m = 36 N ÷ 4 kg = 9 m/s².

    • If you chose B: You multiplied the force by the mass. To find the acceleration, divide the force by the mass.
    • If you chose C: You divided the mass by the force. It is the force divided by the mass.
    • If you chose D: You added the force and the mass. The second law is a division, a = F ÷ m.

  2. A car has a mass of 600 kg. Its engine pulls forward with 2 000 N and resistance pulls back with 500 N, as in the picture. What is its acceleration?

    A car with an arrow to the right labelled drive, 2 000 N and a shorter arrow to the left labelled resistance, 500 N. A bar marked bar = 500 N shows the scale.drive2 000 Nresistance500 Nbar = 500 N
    Tap your answer.

    1. Not this one. You used the engine force alone. The resistance takes some away, so use the resultant. Try another.

    2. Not this one. You added the resistance to the engine force. It pulls the other way, so it takes away. Try another.

    3. Yes, that is right. The resultant is 2 000 N − 500 N = 1 500 N. Then a = 1 500 N ÷ 600 kg = 2.5 m/s².

    See the answer

    2.5 m/s²

    The resultant is 2 000 N − 500 N = 1 500 N. Then a = 1 500 N ÷ 600 kg = 2.5 m/s².

    • If you chose A: You used the engine force alone. The resistance takes some away, so use the resultant.
    • If you chose B: You added the resistance to the engine force. It pulls the other way, so it takes away.

  3. The same resultant force acts on trolley X, of mass 2 kg, and trolley Y, of mass 8 kg. Which accelerates more, and by how much?

    Tap your answer.

    1. Yes, that is right. a = F ÷ m. Trolley Y has 8 ÷ 2 = 4 times the mass, so it has a quarter of the acceleration. Trolley X has 4 times as much.

    2. Not this one. It is the other way round. A bigger mass gives a smaller acceleration for the same force. Try another.

    3. Not this one. The force is the same, but a = F ÷ m. A different mass gives a different acceleration. Try another.

    4. Not this one. The masses are 2 kg and 8 kg. The ratio is 8 ÷ 2, which is 4. Try another.

    See the answer

    Trolley X, 4 times as much.

    a = F ÷ m. Trolley Y has 8 ÷ 2 = 4 times the mass, so it has a quarter of the acceleration. Trolley X has 4 times as much.

    • If you chose B: It is the other way round. A bigger mass gives a smaller acceleration for the same force.
    • If you chose C: The force is the same, but a = F ÷ m. A different mass gives a different acceleration.
    • If you chose D: The masses are 2 kg and 8 kg. The ratio is 8 ÷ 2, which is 4.

  4. A ball of mass 300 g has a resultant force of 3 N on it. What is its acceleration?

    Tap your answer.

    1. Not this one. You put 300 in as the mass. The mass must be in kilograms, so 300 g is 0.3 kg. Try another.

    2. Not this one. You divided the mass by the force. It is the force divided by the mass, in kilograms. Try another.

    3. Yes, that is right. 300 g = 0.3 kg. Then a = F ÷ m = 3 N ÷ 0.3 kg = 10 m/s².

    See the answer

    10 m/s²

    300 g = 0.3 kg. Then a = F ÷ m = 3 N ÷ 0.3 kg = 10 m/s².

    • If you chose A: You put 300 in as the mass. The mass must be in kilograms, so 300 g is 0.3 kg.
    • If you chose B: You divided the mass by the force. It is the force divided by the mass, in kilograms.

  5. Why does a loaded truck speed up more slowly than an empty one with the same engine force?

    Tap your answer.

    1. Not this one. The engine can give the same force. The mass changes, not the force. Try another.

    2. Yes, that is right. By a = F ÷ m, the same force on a bigger mass gives a smaller acceleration.

    3. Not this one. Inertia does not change the force. It changes the acceleration that the force gives. Try another.

    See the answer

    The same resultant force acts on more mass, so the acceleration is smaller.

    By a = F ÷ m, the same force on a bigger mass gives a smaller acceleration.

    • If you chose A: The engine can give the same force. The mass changes, not the force.
    • If you chose C: Inertia does not change the force. It changes the acceleration that the force gives.

  6. The resultant force on an object is doubled. Its mass stays the same. What happens to its acceleration?

    Tap your answer.

    1. Not this one. The acceleration depends on the force. With F = m × a, a bigger force means a bigger acceleration. Try another.

    2. Not this one. Halving happens when the mass is doubled, not the force. Try another.

    3. Not this one. It grows in the same ratio as the force. Double the force makes double the acceleration. Try another.

    4. Yes, that is right. a = F ÷ m. With m fixed, a is in proportion to F. Double the force gives double the acceleration.

    See the answer

    It doubles.

    a = F ÷ m. With m fixed, a is in proportion to F. Double the force gives double the acceleration.

    • If you chose A: The acceleration depends on the force. With F = m × a, a bigger force means a bigger acceleration.
    • If you chose B: Halving happens when the mass is doubled, not the force.
    • If you chose C: It grows in the same ratio as the force. Double the force makes double the acceleration.

  7. From an earlier chapter

    A skater glides across smooth ice at a constant velocity. What is the resultant force on the skater?

    Tap your answer.

    1. Not this one. A forward resultant would make the skater speed up. A constant velocity means no change. Go to S4.8.3. Try another.

    2. Not this one. A backward resultant would slow the skater. The velocity is constant. Go to S4.8.3. Try another.

    3. Yes, that is right. A constant velocity means the acceleration is zero. Then F = m × 0 = 0, so the resultant force is zero.

    See the answer

    It is zero.

    A constant velocity means the acceleration is zero. Then F = m × 0 = 0, so the resultant force is zero.

    • If you chose A: A forward resultant would make the skater speed up. A constant velocity means no change. Go to S4.8.3.
    • If you chose B: A backward resultant would slow the skater. The velocity is constant. Go to S4.8.3.

  8. A resultant force of 72 N gives a trolley an acceleration of 1.5 m/s². What is the mass of the trolley?

    See the answer

    48 kg

    m = F ÷ a = 72 N ÷ 1.5 m/s² = 48 kg.

  9. A crate has a mass of 20 kg. A push of 160 N acts on it and friction pulls back with 20 N. What is its acceleration?

    See the answer

    7 m/s²

    The resultant force is 160 N − 20 N = 140 N. Then a = 140 N ÷ 20 kg = 7 m/s².

Keep it

Copy this into your exercise book.

Second law: resultant force = mass × acceleration, F = m × a. Use the resultant, and the mass in kg. Double the mass and the acceleration halves.

Draw: A box with a big arrow and a small arrow, and the sum a = F ÷ m beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (825 KB). It opens on any phone.