How To Calculate Momentum

From the swinging of a pendulum to a ball rolling down a hill, momentum serves as a useful way to calculate physical properties of objects. You can calculate momentum for every object in motion with a defined mass. Regardless of whether it's a planet in orbit around the sun or electrons colliding with one another at high speeds, the momentum always is the product of the mass and velocity of the object.

You calculate momentum using the equation

\(p=mv\)

where momentum ​p​ is measured in kg m/s, mass ​m​ in kg and velocity ​v​ in m/s. This equation for momentum in physics tells you that momentum is a vector that points in the direction of the velocity of an object. The greater the mass or velocity of an object in motion is, the greater the momentum will be, and the formula applies to all scales and sizes of objects.

If an electron (with a mass of 9.1 × 10 ⁻³¹ kg) was moving at 2.18 × 10⁶ m/s, the momentum is the product of these two values. You can multiply the mass 9.1 × 10 ⁻³¹ kg and the velocity 2.18 × 10⁶ m/s to get the momentum 1.98 × 10 ⁻²⁴ kg m/s. This describes the momentum of an electron in the Bohr model of the hydrogen atom.

You can also use this formula to calculate the change in momentum. The change in momentum ​Δp​ ("delta p") is given by the difference between the momentum at one point and the momentum at another point. You can write this as

\(\Delta p = m_1v_1-m_2v_2\)

for the mass and velocity at point 1 and the mass and velocity at point 2 (indicated by the subscripts).

You can write equations to describe two or more objects that collide with one another to determine how the change in momentum affects the mass or velocity of the objects.

In much the same way knocking balls in pool against one another transfers energy from one ball to the next, objects that collide with one another transfer momentum. According to the law of conservation of momentum, the total momentum of a system is conserved.

You can create a total momentum formula as the sum of the momenta for the objects before the collision, and set this as equal to the total momentum of the objects after the collision. This approach can be used to solve most problems in physics involving collisions.

When dealing with conservation of momentum problems, you consider the initial and final states of each of the objects in the system. The initial state describes the states of the objects just before the collision occurs, and the final state, right after the collision.

If a 1,500 kg car (A) with moving at 30 m/s in the +​x​ direction crashed into another car (B) with a mass of 1,500 kg, moving 20 m/s in the −​x​ direction, essentially combining on impact and continuing to move afterwards as if they were a single mass, what would be their velocity after the collision?

Using the conservation of momentum, you can set the initial and final total momentum of the collision equal to one another as ​p​_Ti = ​p​_Tf​_ ​or ​p_​_A + ​p​_B = ​p​_Tf for the momentum of car A, ​p​_A and momentum of car B, ​p​_B​.​ Or in full, with ​m​_combined as the total mass of the combined cars after the collision:

\(m_Av_{Ai} + m_Bv_{Bi} = m_{combined}v_f\)

Where ​v​_f is the final velocity of the combined cars, and the "i" subscripts stand for initial velocities. You use −20 m/s to for the initial velocity of car B because it's moving in the −​x​ direction. Dividing through by ​m​_combined (and reversing for clarity) gives:

\(v_f = \frac{m_Av_{Ai} + m_Bv_{Bi}}{ m_{combined}}\)

And finally, substituting the known values, noting that ​m​_combined is simply ​m​_A + ​m​_B, gives:

\(\begin{aligned}
v_f &= \frac{1500 \text{ kg} × 30 \text{ m/s} + 1500 \text{ kg} ×-20 \text{ m/s}}{ (1500 + 1500) \text{ kg}} \
&= \frac{45000 \text{ kg m/s} – 30000 \text{ kg m/s}}{3000 \text{ kg}} \
&= 5 \text{ m/s}
\end{aligned}\)

Note that despite the equal masses, the fact that car A was moving faster than car B means the combined mass after the collision continues to move in the +​x​ direction.