# Impulse and Momentum Calculator

Created by Davide Borchia
Last updated: Aug 16, 2022

Discover how time, mass, and velocity affect the outcome of a collision with our impulse and momentum calculator. We will guide you on an interesting journey in simple physics with significant consequences: discover the equations that control the change in the momentum of a body.

• What is momentum, and why do we need it when describing the motion of a body;
• What happens when we change momentum: the impulse, its formula, and its relation with velocity;
• How to calculate impulse in some more-or-less realistic examples. Bullets and nearly supersonic baseball balls;
• The impulse-momentum equation: adding time to the equation — literally.

## Just a momentum: the importance of mass in kinematics

The description of the motion of an object is rarely complete without including its mass. While there are instances where it is possible to neglect this quantity — and demonstrate important truths of our physical world, as in free-fall motions —, the mass of a moving body is of prime relevance when collisions, external forces, and non-idealities of the real world appears in our problems.

Momentum is the first fundamental quantity that sees the mass of a body considered during the study of its motion. In physics, the formula for momentum is:

$\boldsymbol{p}= m\cdot\boldsymbol{v}$

Where:

• $\boldsymbol{p}$ is the momentum;
• $m$ is the mass; and
• $\boldsymbol{v}$ is the velocityof the object.

Both the velocity and the momentum are vectorial quantities.

## A change in momentum: the formula for the impulse

Everyone has an intuitive understanding of momentum: while we don't flinch if someone throws a pop-corn at us, we try to dodge an incoming pebble. The key is in the difference in mass.

An impact of a body with a higher momentum is more energetic than the one of a body with a lower momentum: the forces involved are bigger.

The magnitude of these forces is directly related to the change in momentum. We can slightly change our perspective and think of the force as the cause of the change in momentum: the wall against which you threw that ball was the one stopping the ball, and not the ball hit the wall with a certain force.

Within this framework, we can easily see how a force is directly connected to a change in momentum. The change in momentum, nominally $\boldsymbol{\Delta p}$ has, of course, the same units of momentum. We call this quantity the impulse. The impulse has a formula:

$\boldsymbol{J} = \boldsymbol{\Delta p} = \boldsymbol{p}_{\text{f}} - \boldsymbol{p}_\text{i}$

Where:

• $\boldsymbol{J}$ is the impulse;
• $\boldsymbol{p}_\text{i}$ is the initial momentum; and
• $\boldsymbol{p}_\text{f}$ is the final momentum.

Going just a bit further, we can see how the impulse is related to the change in velocity of the body:

$\boldsymbol{J} =m\cdot (\boldsymbol{v}_{\text{f}} - \boldsymbol{v}_\text{i})$

Now that you know how to calculate the impulse let's apply the math to an example. Imagine a collision between a massive body and a projectile. Maybe a cal. .50 BMG bullet hitting a bulletproof glass (possibly mounted on sturdy support since the guns firing those things are called anti-materiel for a reason). The momentum (rather big) of the projectile (with mass $m=43\ \text{g}$) traveling at $2.5$ times the speed of sound would hopefully be reduced to $0$, stopping the round before penetration. Let's applicate the impulse equation in this situation:

\begin{align*} \boldsymbol{J} &=43\cdot 10^{-3}\ \text{kg}\cdot \left(0-854\frac{\text{m}}{\text{s}}\right)\\ \\ &=-36.7\ \text{N}\cdot\text{s} \end{align*}

Take a baseball ball with mass $145\ \text{g}$. A good throw speed is $\boldsymbol{v}=92\ \text{mph} = 41.1\ \text{m}/\text{s}$. How fast would the ball go if we applied the same impulse as the above example?

You can insert the following quantities in our impulse and momentum calculator:

• Impulse $-36.7$ (the minus sign is fundamental!);
• Mass $145$ (remember to select grams as unit); and
• Initial velocity $41.1$.

With mass and impulse, our impulse and momentum calculator can quickly compute the change in velocity. At this point, the ball's final velocity is nothing but simple math. From the formula of the change in momentum:

\begin{align*} \boldsymbol{v}_{\text{f}} &= \frac{\boldsymbol{J}}{m}-\boldsymbol{v}_\text{i}\\ \\ &=\frac{-36.7\ \text{N}\cdot\text{s}}{0.145\ \text{kg}}-41.1\ \frac{\text{m}}{\text{s}}\\ \\ &=-212\ \frac{\text{m}}{\text{s}} = 474.2\ \text{mph} \end{align*}

A bit slower than a passenger airliner. You'd better not catch that ball!

## Derive the impulse equation from Newton's second law

The physics of impulse doesn't stop here. Do you remember Newton's second law? Here it is, in all its derivative glory:

$\boldsymbol{F} = \frac{d\boldsymbol{p}}{d t}$

Look at it: the right hand side is, if we consider bigger variations, the change in momentum $\boldsymbol{\Delta p}$ in a certain time $\Delta t$.

Slightly rearrange the equation above:

$\boldsymbol{F}\cdot \Delta t = \boldsymbol{\Delta p}$

Compare this result to the formula for the impulse we found in the previous section: you will find that $\boldsymbol{J} = \boldsymbol{F}\cdot \Delta t$. The last result is also known as the impulse-momentum equation and — rather elegantly — explains the relationship between force, time, and change in momentum.

A large force distributed over a relatively short time has the same impulse as a weak force applied for longer. The relative change in momentum is the same.

The importance of time during a change in momentum is not to underestimate. The physics of impulse determines the usefulness of bulletproof glass: by increasing the time of the impact, they drastically reduce the force applied to the body.

Remember the cal. 50 BMG we fired before? The bulletproof glass (or vest) we shoot against has the sole purpose of prolonging the impact duration. The longer the time, the smaller the force, and the smaller the damage.

On the contrary, smaller times are desirable when the available force is not that big. In golf, boxing, and many other sports the player tries to reduce the time of contact to deliver the strongest possible hit.

Davide Borchia
Velocity change (Δv)
ft/s
Initial velocity (v₁)
ft/s
Final velocity (v₂)
ft/s
Mass (m)
lb
Force (F)
N
Time interval
sec
Impulse and momentum
Impulse (J)
N·s
Initial momentum (p₁)
N·s
Final momentum (p₂)
N·s
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