# Momentum Calculator

Created by Krishna Nelaturu
Reviewed by Dominik Czernia, PhD candidate and Steven Wooding
Last updated: Dec 01, 2021

Our linear momentum calculator helps you calculate the momentum of an object in linear motion. A moving object possesses momentum by virtue of its mass and velocity. If the motion is linear (translational), the object has linear momentum, and if the motion is rotational, the object has angular momentum.

Do you want to know the formula for momentum, its units, and how to calculate the magnitude of momentum in two or three dimensions? Are you curious about how it is related to dear old Newton’s second law of motion? Look no further because you’ve come to the right place!

## How do you use this momentum calculator?

You can use our momentum calculator to:

1. Calculate momentum using mass and velocity: Enter the values for mass and velocity in desired units to calculate momentum.
2. Find mass using momentum and velocity or velocity using momentum and mass: Enter the values of momentum and velocity (or mass) in desired units. Our tool will inverse the momentum formula and find the missing value.
3. Determine the components of momentum and its magnitude in two or three dimensions using advanced mode: Click on Advanced mode at the bottom of the momentum calculator, and enter the values of mass and velocity components to calculate momentum components and momentum magnitude, or vice versa.

🙋 Tip: If you want to set the value of a variable, say mass, as a constant throughout your calculations, then click on the right end of the variable field to open the row menu and select lock.

## What is momentum of an object? How to calculate momentum

Linear momentum (or simply momentum) is the product of an object’s mass with its velocity. We denote it by $\mathbf{p}$:

\qquad \begin{align*} \mathbf{p}= m\mathbf{v} \end{align*}

where:

• $\mathbf{p}$ – Momentum of the object;
• $m$ – Mass of the object; and
• $\mathbf{v}$ – Velocity of the object.

The SI units of momentum are kg·m/s or N·s, and imperial units are lbs·ft/s. Since mass is a scalar and velocity is a vector, momentum is also a vector quantity, with the same direction as the velocity.

Let us consider how to calculate momentum with some examples:

• A person weighing $65 \text{ kg}$, jogging at $2 \text{ m/s}$ has a momentum magnitude of $65\text{ kg} \cdot 2\text{ m/s}=130 \text{ N⋅s}$. If they want to have a higher momentum, say $195 \text{ N⋅s}$, then they would have to jog at $\frac{195}{65} = 3\text{ m/s}$ by exerting more effort (or force).

• An empty dump truck weighing $12 \text{ t}$ travelling at $35 \text{ mph}$ has a momentum of $12 \text{ t} \cdot 35\text{ mph} = 420 \text{ t⋅mph}$ or 187.8 kN⋅s. If the dump truck is loaded to its full capacity of $12 \text{ t}$, then its weight would be doubled, and hence so would be it's momentum.

• A charging elephant can run at a speed of $35 \text{ km/h}$. It may not seem like much, but since they can weigh anywhere between $2 \text{ t}$ to $7 \text{ t}$, they can gain tremendous momentum of about $70 \text{ t⋅km/h}$ (19.4 kN⋅s) to $245 \text{ t⋅km/h}$ (68.1 kN⋅s) in their charge. So you can see why it’s wiser to get out of dodge.

## Linear momentum equation in two and three dimensions

We've established that momentum is a vector. If velocity has non-zero components along more than one axis in the selected frame of reference, so shall momentum. Linear momentum equation in vector form:

\begin{align*} \mathbf{p} = \begin{bmatrix} p_x \\ p_y \\ p_z \end{bmatrix} = m\mathbf{v} = m\begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}\\ \end{align*}

From this vector equation, we can extract the momentum formula for each component:

\qquad \begin{align*} p_x &= mv_x \\ p_y &=mv_y \\ p_z &=mv_z \end{align*}

Once again, it is evident that the direction of the momentum vector is along the direction of the velocity. You can calculate these components in our momentum calculator by selecting advanced mode.

It follows that the total magnitude of momentum is:

\begin{align*} \Vert \mathbf{p}\Vert &=\sqrt{p_x^2+p_y^2+p_z^2}\\ &=m\sqrt{v_x^2+v_y^2+v_z^2}\\ \implies\Vert \mathbf{p}\Vert &= m\Vert \mathbf{v}\Vert \end{align*}

In other words, the magnitude of momentum is the product of mass and magnitude of velocity.

Use our vector magnitude calculator to find the magnitude of vectors in even five-dimensional space!

## Momentum examples: Why is momentum important?

Equations and knowing how to calculate momentum are great, but is there any significance to momentum, you ask. The following is a brief discussion of how understanding the momentum of a system can be useful:

• Let us revisit the dump truck momentum example from earlier. The empty dump truck moving at 35 mph is now, say, approaching its destination. To stop the truck, the driver has to decelerate the vehicle by applying force in the form of brakes. The driver can use a large amount of brake force to stop the truck abruptly or brake over time to decelerate to rest smoothly. This case (and other observations) shows that the rate of change of momentum is directly proportional to the force applied to the system, as expressed in Newton's second law of motion.

• In some cases, a large force acting in a short period can change the object's momentum. Consider the case of a ball bouncing off of a wall – initially, the ball has a finite momentum, but upon contact with the wall, its momentum changes direction in the short interval of contact. It is usually impossible to determine such short periods and large forces, but the change in momentum remains measurable. We call a large force acting on a body in a short period an impulse.

• Law of conservation of momentum states that when no external forces are acting on a system, the total momentum of all the objects in the system remains unchanged. In other words, individual objects of the system may interact and change each other's momentum. However, so long as no external forces are introduced, the changes in momentum negate each other, and the total momentum of the system is conserved.

Krishna Nelaturu
Momentum in one dimension
Mass (m)
lb
Velocity (v)
ft/s
Momentum (p)
kg⋅m/s
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