Newton’s Second Law of Motion
Force saves you from the monotony of everything moving at the same speed and direction forever. Force can act on objects, changing their direction and/or speed. The relationship between force, mass, and acceleration is primary in physics classes.
To start, you need to know Newton’s Second Law of Motion, which is a big one in physics: “When a net force sigmaF acts on an object of mass m, the acceleration of that mass can be calculated by sigmaF = ma.” The sigma sign stands for “sum,” so sigmaF = ma can be read as “the vector sum of all forces on an object (the net force) equals mass times acceleration.” It is important to note that this is a vector equation and that both force and acceleration are vectors in this equation. Mass, like time, is a scalar.
Newton’s second law is often just abbreviated as F = ma. It’s important to note that sigmaF refers to the vector sum of all the forces on the object and not to just a single force.
What is the unit of force? This table gives you a rundown for the three measurement systems most commonly used in physics:
System  Unit  Name  Abbreviation 

MKS  kgm/s^{2}  newton  N 
CGS  gm/s^{2}  dyne  dyn 
FPS  pound  lb 
Here’s how you relate the three different units of force:
1 lb = 4.448 N
1 N = 10^{5} dyn
Sample question

You’re at rest on an ice rink when you get hit from behind with a force of 50.0N as someone bumps you. If your mass is 70.0 kg, what is your acceleration?
The correct answer is 0.714 m/s^{2}.

Use the equation F = ma: Solving for a gives you a = F/m.

Plug in the numbers: a = F/m = (50.0 N)/(70.0 kg) = 0.714 m/s^{2}.

Practice questions

You come home to find a delivered package with a mass of 100 kg blocking the door. If you push it with a force of 100 N, what will its acceleration be if no friction is involved?

You’re gliding across a frictionless lake in a sailboat. If your mass is 70 kg and the boat’s mass is 200 kg, with what force does the wind need to blow you to give you an acceleration of 0.30 m/s^{2}?

You have control of a space station, which has a mass of 400,000 kg. To give it an acceleration of 2.0 m/s^{2}, what force do you need to apply with the rockets?

You find a stone in the forest and give it a push of 50.0 N. It accelerates at 2.0 m/s^{2}. What is its mass?

You’re applying a force of 17 N to a hockey puck with a mass of 0.17 kg. Starting from rest, how far has the puck gone in 0.10 seconds?

You push a rowboat on a calm lake (assuming no friction) with a force of 40.0 N. If the rowboat has a mass of 80.0 kg, how far has it gone in 10.0seconds?

A space station with a mass of 1.0 x 10^{5} kg is moving toward a satellite at 5.0 m/s. If you want to avoid crashing them together and have only 1.0 x 10^{3} seconds in which to act, what force do you need to apply to stop the space station from colliding with the satellite?

Your 1,000kg car needs a push. Starting at rest, how hard do you have to push to get it up to 5.0 m/s in 1.0 x 10^{2} seconds?
Following are answers to the practice questions:

1 m/s^{2}

Solving F = ma for a gives you a = F/m.

Plug in the numbers: a = F/m = 100 N/100 kg = 1 m/s^{2}.


81 N

Use the equation F = ma.

Plug in the numbers: F = ma = (70 kg + 200 kg)(0.30 m/s^{2}) = 81 N.


800,000 N

Use the equation F = ma.

Plug in the numbers: F = ma = (400,000 kg)(2.0 m/s^{2}) = 800,000 N.


25 kg

Use the equation F = ma, and solve for the mass, giving you m = F/a.

Plug in the numbers: m = F/a = 50 N/2.0 m/s^{2} = 25 kg.


50 m

Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

Use the equation
and substitute F/m for a:

Plug in the numbers:


25 m

Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

Use the equation
and substitute F/m for a:

Plug in the numbers:


–5.0 x 10^{2} N

Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

Use the equation v_{f} = v_{o} + at. In this question, v_{f} = 0, so at = –v_{o}.

This becomes
solving for F gives you

Plug in the numbers, and you get F = –500 N = –5.0 x10^{2} N (it’s negative because it’s opposite to the direction of travel).


50 N

Use the equation F = ma, and solve for the acceleration, giving you a = F/m.

Use the equation v_{f} = v_{o} + at. In this question, v_{f} = 10 and v_{o} = 0, so at = v_{f}.

This becomes
solving for F gives you

Plug in the numbers, and you get F = 50 N.
