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Newton's second law: F = ma

Key idea: Resultant force equals mass times acceleration. Acceleration points in the resultant force's direction; the same force gives a smaller acceleration to a larger mass.

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Push an empty shopping trolley and a full one with the same force, and the empty one speeds up far more quickly. A resultant force makes an object accelerate, and one equation says by how much: resultant force = mass × acceleration. With it you can find an acceleration, a mass or a missing force, once you have found the resultant force with its direction.

Resultant force = mass × acceleration

F = ma
SymbolQuantitySI unit
Fresultant forcenewton (N)
mmasskilogram (kg)
aaccelerationmetre per second squared (m/s²)

One newton is the resultant force that gives a 1 kg mass an acceleration of 1 m/s². The acceleration is always in the direction of the resultant force.

The relationship says two things:

  • for the same mass, a larger resultant force gives a larger acceleration;
  • for the same resultant force, a larger mass gives a smaller acceleration.
Effect of mass on acceleration for the same forceThe same horizontal push acts on a pail containing less sand and an otherwise identical pail containing more sand. The lighter pail has greater acceleration and the heavier pail has smaller acceleration.same Flarger accelerationless sand: smaller masssame Fsmaller accelerationmore sand: larger mass
Effect of mass on acceleration for the same forceThe same horizontal push acts on a pail containing less sand and an otherwise identical pail containing more sand. The lighter pail has greater acceleration and the heavier pail has smaller acceleration.same Flarger accelerationless sand: smaller masssame Fsmaller accelerationmore sand: larger mass
The same resultant force on two pails: the pail with less sand, and so less mass, has the larger acceleration.

Check your understanding 1 Two trolleys, one force

The same resultant force acts on a 2.0 kg trolley and on a 6.0 kg trolley. Compare their accelerations.

Show answer

a = F/m, so the 2.0 kg trolley’s acceleration is three times the 6.0 kg trolley’s.

Rearranged, a = F/m and m = F/a. In every case, F is the resultant force, found from the free-body diagram, not just the force someone applies.

A method for every F = ma problem

  1. Draw the free-body diagram of the body.
  2. Choose a positive direction.
  3. Find the resultant force, with its sign.
  4. Use F = ma with mass in kilograms.
  5. State the answer with its unit and direction. A negative acceleration points in the negative direction.

Worked example 1

Acceleration of a pushed trolley

Problem

A 4.0 kg trolley is pushed to the right with 18 N. Friction on it is 6.0 N to the left. Find its acceleration.

Show full solution

  1. Choose directions

    Method

    Take right as positive and use the horizontal forces only.

    Reason

    The vertical forces balance, so they do not affect the horizontal motion.

    Working

    Answer

    Right = +

  2. Find the resultant force

    Method

    Subtract friction from the push.

    Reason

    Friction acts to the left, against the chosen direction.

    Working

    Answer

    F = 18 - 6.0 = 12 N

  3. Apply F = ma

    Method

    Divide the resultant by the mass.

    Reason

    Acceleration depends on the resultant force.

    Working

    Answer

    a = F/m = 12/4.0 = 3.0 m/s²

  4. State and check the answer

    Working

    Answer

    The acceleration is 3.0 m/s² to the right. Using the 18 N push alone would give 4.5 m/s², too large because friction also acts.

Check your understanding 2 A rising drone

A 3.0 kg drone has 45 N of upward thrust, a weight of 30 N and 3.0 N of air resistance acting downward. Take upward as positive. Find the resultant force and acceleration.

Unit: N
Show the steps Optional working
Unit: N
Unit: m/s²
Show answer

Take upward as positive. Resultant force: F = 45 - 30 - 3.0 = 12 N upward. Then a = F/m = 12/3.0 = 4.0 m/s² upward.

Worked example 2

A braking car

Problem

A 1200 kg car moves to the right. The driver brakes, and the braking force on the car is 6000 N to the left. Ignore the other horizontal forces. Find the car’s acceleration.

Show full solution

  1. Choose directions

    Method

    Take right, the direction of motion, as positive.

    Reason

    A sign convention shows whether the car speeds up or slows down.

    Working

    Answer

    Right = +

  2. Find the resultant force

    Method

    Give the braking force its sign.

    Reason

    The braking force acts to the left.

    Working

    Answer

    F = -6000 N

  3. Apply F = ma

    Method

    Divide the resultant by the mass.

    Reason

    Acceleration has the direction of the resultant force.

    Working

    Answer

    a = (-6000)/1200 = -5.0 m/s²

  4. Interpret and check

    Working

    Answer

    The acceleration is to the left, opposite to the velocity, so the car slows down: its deceleration is 5.0 m/s². At 20 m/s it would stop in 20 ÷ 5.0 = 4.0 s, sensible for hard braking.

Check your understanding 3 Freewheeling

A cyclist and bicycle have a total mass of 60 kg. The cyclist stops pedalling, and 150 N of resistance acts against the motion. Take forward as positive. Find the acceleration and choose the immediate motion.

Unit: m/s²
Immediate motion
Show answer

Take forward as positive: a = F/m = -150/60 = -2.5 m/s². The acceleration is backward, so the cyclist slows down at 2.5 m/s².

Worked example 3

Finding an unknown force

Problem

A 6.0 kg box has a 22 N force to the right and one other horizontal force acting on it. It accelerates at 2.5 m/s² to the left. Find the other force.

Show full solution

  1. Give the acceleration its sign

    Method

    Take right as positive.

    Reason

    The acceleration is to the left.

    Working

    Answer

    a = -2.5 m/s²

  2. Find the resultant force

    Method

    Use F = ma.

    Reason

    The resultant force causes the given acceleration.

    Working

    Answer

    F = 6.0 × (-2.5) = -15 N, i.e. 15 N to the left

  3. Solve for the other force

    Method

    Call the other force P, to the left, and add the forces.

    Reason

    The forces must add to the resultant.

    Working

    Answer

    22 - P = -15, so P = 37 N

  4. State and check the answer

    Working

    Answer

    The other force is 37 N to the left. Check: 22 - 37 = -15 N, and -15 ÷ 6.0 = -2.5 m/s², as given.

Check your understanding 4 Finding a mass

A resultant force of 24 N gives a trolley an acceleration of 1.5 m/s². Find the trolley’s mass.

Unit: kg
Show answer

m = F/a = 24/1.5 = 16 kg.

Check your understanding 5 Use observed motion to recover an unknown force

A 500 g cart moves right and slows uniformly from 8.0 m/s to 4.0 m/s in 2.0 s. A motor still pushes it right with 0.20 N. The only other horizontal force is resistance to the left. Find the signed acceleration with right positive, the resultant force and the resistive force.

Unit: kg
Show the steps Optional working
Unit: m/s
Unit: m/s²
Show the steps Optional working
Unit: kg
Unit: m/s²
Unit: N
Show the steps Optional working
Unit: N
Unit: N
Show answer

Mass = 0.500 kg. Acceleration = (4.0 − 8.0) ÷ 2.0 = −2.0 m/s², directed left. Resultant force = 0.500 × (−2.0) = −1.0 N. If resistance has magnitude R, 0.20 − R = −1.0, so R = 1.20 N to the left. The motor alone would give a rightward acceleration and speeding up; the observed slowing requires the larger opposing force. Check: (0.20 − 1.20) ÷ 0.500 = −2.0 m/s².

Common mistakes

  • Putting the applied force into F = ma instead of the resultant force.
  • Dropping the minus sign in a braking problem. With forward taken as positive, negative acceleration slows a body moving forwards.
  • Using a mass in grams. Convert first: 500 g = 0.50 kg.
  • Using a weight in newtons in place of the mass. Use W = mg, with g = 10 N/kg, to find the mass.
  • Drawing ma as another arrow on the free-body diagram. It is what the real forces add up to.

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