Newton's second law is the relationship that lets you predict motion. It says that how quickly an object's velocity changes depends on two things: the net force acting on it and its mass. In Physics you analyze data to support this claim for motion in one dimension, and you use the law to calculate a force, a mass or an acceleration when the other two are known.
The law
a = Fnet ÷ m, usually written Fnet = m · a.
- Fnet is the vector sum of all forces on the object, in newtons (N).
- m is the object's mass, in kilograms (kg).
- a is its acceleration, in meters per second squared (m/s²).
One newton is the force that gives a 1 kg mass an acceleration of 1 m/s², so 1 N = 1 kg·m/s². The acceleration points in the same direction as the net force. Two proportions follow: for a fixed mass, doubling the net force doubles the acceleration, and for a fixed force, doubling the mass halves the acceleration.
What the data look like
Suppose you pull carts of different masses with the same 10 N net force and measure the acceleration each time.
| Mass (kg) | Net force (N) | Acceleration (m/s²) |
|---|---|---|
| 1.0 | 10 | 10 |
| 2.0 | 10 | 5.0 |
| 5.0 | 10 | 2.0 |
Acceleration times mass is 10 N every time. That constant product is the evidence for the law. A graph of acceleration against 1/m is a straight line through the origin.
Worked example 1: find the acceleration
A 1500 kg car has a net forward force of 3000 N. What is its acceleration?
a = F ÷ m = 3000 N ÷ 1500 kg = 2.0 m/s².
If the car starts from rest and the force stays the same for 5.0 s, its speed is 2.0 m/s² × 5.0 s = 10 m/s.
Worked example 2: find the net force first
A 20 kg crate is pushed to the right with 60 N, while friction pushes left with 20 N. What is its acceleration?
- Net force = 60 N − 20 N = 40 N to the right.
- a = 40 N ÷ 20 kg = 2.0 m/s² to the right.
The law always uses the net force. Using only the 60 N would give 3.0 m/s², which is wrong.
Worked example 3: mass and weight
Weight is the gravitational force on an object, so it is just the second law with a = g, where g is about 9.8 m/s² near Earth's surface. For a 70 kg person:
W = m · g = 70 kg × 9.8 m/s² = 686 N, which rounds to 6.9 × 10² N with two significant figures.
The person's mass is 70 kg anywhere. Their weight changes with g. On the Moon, where g is about 1.6 m/s², the weight is about 1.1 × 10² N, while the mass is still 70 kg.
Using the law to find an unknown
Rearrange to suit what you are missing: m = F ÷ a, or F = m · a. If a 0.50 kg cart accelerates at 4.0 m/s², the net force is 0.50 kg × 4.0 m/s² = 2.0 N. If the same 2.0 N acts on a 1.0 kg cart, the acceleration is 2.0 ÷ 1.0 = 2.0 m/s², half as much.
Significant figures and units
Give a result no more significant figures than the least precise measurement. Always write the unit. Before you calculate, convert grams to kilograms and any other unit to the SI ones above. A quick unit check helps: N ÷ kg = (kg·m/s²) ÷ kg = m/s², so the unit comes out right.
Worked example 4: braking
A 0.80 kg hockey puck slides on ice and slows at 0.50 m/s². What friction force acts on it? F = m · a = 0.80 kg × 0.50 m/s² = 0.40 N, directed opposite to the motion. The same push of 0.40 N would speed up the puck at the same rate if it pointed the other way.
Common mistakes
- Using one force instead of the net force. Add all the forces, with direction, first.
- Confusing mass and weight. Mass is in kg and does not change. Weight is a force in newtons.
- Thinking a force is needed to keep an object moving. Zero net force means constant velocity, not zero velocity.
- Leaving mass in grams. 500 g is 0.500 kg.
- Ignoring direction. Acceleration is a vector. A negative sign means it points the opposite way from the one you chose as positive.
Practice Physics questions with an explanation for every answer, or read the guide to California Physics practice.
Source: the California Next Generation Science Standards (CA NGSS), as set out in the California Science Framework, HS-PS2-1.