Momentum and forces

There’s a special link between the momentum of an object and the forces which act on the object.

If we apply a force of F newtons to an object, for t seconds, the change in momentum of the object (the impulse) is the force multiplied by the time for which the force is applied:

\text{impulse} = F \times t

Change in momentum

If we replace the word impulse with the change in momentum, we can write the equation like this:

\Delta p = F \times t

The full equation

We know that the equation for momentum is:

p = mv

That means we can replace p for momentum with mv in the first equation here:

\Delta mv = F \times t

You may see this rewritten as:

F = \frac{\Delta mv}{t}

Where:

• F is the force in newtons (N)
• \Delta mv is the change in momentum in kilogram meters per second (kgms^{-1}). Otherwise known as the impulse.
• t is the time for which the force is applied in seconds (s)

Force-time graphs

If we plot a graph with force on the y-axis and time on the x-axis:

• The y-axis will show the force, in Newtons (N)
• The x-axis will show the time, in seconds (s)
• We know that one Newton is one kilogram meter per second squared (N = kgms^{-2}) because a Newton is the force required to accelerate a mass of one kilogram at a rate of one meter per second squared.
• So if we multiply the y-axis (force) by the x-axis (time), we get kilogram meters per second (kgms^{-1}), which is the unit for momentum!

That shows that:

The area under a force-time graph is equal to the change in momentum (or impulse) of the object.

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