Lesson Video: Linear Momentum Mathematics

Video Transcript

In this video, we will learn how to calculate the momentum of a particle moving in a straight line using the formula 𝐇 equals 𝑚𝐯. Imagine two objects, a truck moving at 30 miles per hour along a road and a paper aeroplane moving at two miles per hour through the air. Which object would require a greater force to stop it in the same amount of time?

Intuitively, we know that the truck would require the greater force to stop it because it has a greater mass and it is moving faster. We can therefore say that the truck has a greater momentum. Momentum can be thought of as a measure of how difficult it is to stop an object that is moving.

Let’s begin by considering a more formal definition of this. The two factors that contribute to an object’s momentum are its mass 𝑚 and its velocity vector 𝐯. The greater the mass of the object, the greater its momentum. And in the same way, the greater the velocity of the object, the greater its momentum. We can therefore define the momentum of an object vector 𝐇 as equal to its mass 𝑚 multiplied by its velocity vector 𝐯. This is written 𝐇 is equal to 𝑚𝐯.

Since velocity is a vector quantity and mass is a scalar quantity, momentum is a vector quantity. However, we often just want the magnitude of the momentum. We can therefore write that the magnitude of vector 𝐇 is equal to the magnitude of 𝑚𝐯. And since mass is a scalar quantity, we can take it outside of the magnitude sign, giving us the magnitude of vector 𝐇 is equal to 𝑚 multiplied by the magnitude of vector 𝐯. On the right-hand side, the magnitude of vector 𝐯 is simply the magnitude of the velocity or the speed. We can denote the magnitude of the momentum just as 𝐻 and the speed as just 𝑣. This gives us 𝐻 is equal to 𝑚𝑣.

If we consider a bowling ball with a mass of 12 kilograms moving at a speed of five meters per second along the lane of a bowling alley, we can calculate the momentum of the bowling ball by substituting the values into the formula 𝐻 equals 𝑚𝑣. The momentum 𝐻 is equal to 12 kilograms multiplied by five meters per second. As 12 multiplied by five is equal to 60, the momentum is equal to 60 kilogram meters per second. This shows us that the standard unit of momentum is kilogram meters per second. However, momentum can also be measured in other units, in fact any unit of mass multiplied by a unit of speed.

We will now look at some examples.

Determine the momentum of a car of mass 2.1 metric tons moving at 42 kilometers per hour.

We begin by recalling that the two factors that affect an object’s momentum are its mass and speed such that the momentum 𝐻 is equal to the mass 𝑚 multiplied by the speed 𝑣. In this question, we are told the mass of the car is 2.1 metric tons and its speed is 42 kilometers per hour. Substituting these values into the formula, we see that 𝐻 is equal to 2.1 tons multiplied by 42 kilometers per hour. To calculate 2.1 multiplied by 42, we can multiply two by 42 and then 0.1 by 42. These are equal to 84 and 4.2, respectively. As these two values sum to give us 88.2, 2.1 multiplied by 42 is equal to 88.2. We can therefore conclude that the momentum of the car is 88.2 ton kilometers per hour.

This is a slightly unusual unit as the standard unit of momentum is kilogram meters per second. However, any unit of mass multiplied by a unit of speed is a valid unit of momentum. Had we been required to give our answer in kilogram meters per second, we could’ve converted the mass to kilograms and the speed to meters per second before multiplying our values.

In our next example, we will calculate the momentum by first using the equations of motion.

Calculate the momentum of a stone of mass 520 grams after it has fallen 8.1 meters vertically downward. Consider the acceleration due to gravity to be 𝑔, which is equal to 9.8 meters per second squared.

In this question, we’re not given the speed of the stone, which we need to know in order to calculate the momentum. We are told how far the stone falls and what its acceleration is. As we are not told anything about the initial motion of the stone, we can assume that initially it was at rest.

Knowing these three pieces of information means we can use the equations of motion or SUVAT equations to help calculate the speed of the particle after it has fallen 8.1 meters. We know that displacement 𝑠 is 8.1 meters. The initial velocity 𝑢 is zero meters per second. We are trying to calculate the final velocity 𝑣 in meters per second. And we are told the acceleration due to gravity is 9.8 meters per second squared.

In this question, we know nothing about the time. We can therefore use the equation 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values of 𝑢, 𝑎, and 𝑠, we have 𝑣 squared is equal to zero squared plus two multiplied by 9.8 multiplied by 8.1. 𝑣 squared is therefore equal to 158.76. Square rooting both sides of this equation and recognizing that the speed must be positive, we have a value of 𝑣 equal to 12.6. After the stone has fallen 8.1 meters vertically downwards, it is traveling at a speed of 12.6 meters per second.

Next, we recall that the momentum of any particle 𝐻 is equal to its mass 𝑚 multiplied by its speed 𝑣. The standard units of momentum are kilogram meters per second, as the standard units of mass are kilograms and speed are meters per second. We are told that the mass of the stone is 520 grams, and we know there are 1,000 grams in a kilogram. This means that the mass of the stone in kilograms is 0.52.

We can therefore calculate the momentum of the stone by multiplying 0.52 by 12.6. Typing this into the calculator gives us 6.552. We can therefore conclude that by the time the stone has fallen a distance of 8.1 meters, it had a momentum of 6.552 kilogram meters per second.

In our next question, we will calculate the change in momentum of a particle given its acceleration and initial velocity.

A body of mass 17 kilograms moves in a straight line with constant acceleration of 1.8 meters per second squared. Its initial velocity is 22.3 meters per second. Find the increase in its momentum in the first five seconds.

This question asks us to find the increase in momentum or change in momentum over a given time period. In order to do this, we need to work out the difference between its final momentum and its initial momentum. We can express this mathematically as Δ𝐻 is equal to 𝐻 sub two minus 𝐻 sub one, where Δ𝐻 is the change in momentum, 𝐻 two is the final momentum, and 𝐻 one is the initial momentum. As momentum is equal to mass multiplied by speed, this can be rewritten as Δ𝐻 is equal to 𝑚𝑣 sub two minus 𝑚𝑣 sub one, where 𝑣 sub two is the final speed, 𝑣 sub one is the initial speed, and 𝑚 is the mass of the body.

We are told in the question the initial speed of the body is 22.3 meters per second. We also know that the mass is 17 kilograms. This means we need to begin by calculating the final speed. We can do this using our equations of motion, sometimes known as the SUVAT equations.

We know that the initial velocity is 22.3 meters per second. The acceleration 𝑎 is 1.8 meters per second squared. And the time period we are interested in is five seconds. We can therefore calculate the final velocity 𝑣 using the equation 𝑣 is equal to 𝑢 plus 𝑎𝑡. Substituting in the values of 𝑢, 𝑎, and 𝑡, we have 𝑣 is equal to 22.3 plus 1.8 multiplied by five. 1.8 multiplied by five is equal to nine. Therefore, 𝑣 is equal to 31.3. The velocity of the body after five seconds is 31.3 meters per second.

We now have values of 𝑚, 𝑣 sub one, and 𝑣 sub two. We know the initial speed, final speed, and the mass of the object. The change in momentum is therefore equal to 17 multiplied by 31.3 minus 17 multiplied by 22.3. Whilst we could type this straight into our calculator, we notice that the mass 𝑚 is common to both terms on the right-hand side. We can therefore rewrite the change in momentum as 𝑚 multiplied by 𝑣 sub two minus 𝑣 sub one.

In this question, the change in momentum is equal to 17 multiplied by 31.3 minus 22.3. This simplifies to 17 multiplied by nine, which in turn is equal to 153. As the mass of the body was measured in kilograms and the speed or velocities were in meters per second, we use the standard units of momentum of kilogram meters per second. In the first five seconds of motion, the momentum of the object increases by 153 kilogram meters per second.

In our final question, we will calculate the momentum given an equation for the particle’s displacement and the mass of the particle.

A car of mass 1,350 kilograms moves in a straight line such that at time 𝑡 seconds, its displacement from a fixed point on the line is given by 𝑠 is equal to six 𝑡 squared minus three 𝑡 plus four meters. Find the magnitude of the car’s momentum at 𝑡 equals three seconds.

In this question, we are given a function for the position of the car that depends only on time. In order to find the momentum of the car at a particular time, we are going to need to know its velocity at that time. In order to get this velocity, we will firstly need to get a general function for the velocity of the car over time.

We recall that the velocity of an object vector 𝐯 is defined as the rate of change of the displacement of the object. It is therefore the derivative of the displacement of the object with respect to time. Since this question is just about an object moving in one dimension, we can use scalar quantities to represent the velocity 𝑣 and the displacement 𝑠, giving us 𝑣 is equal to d𝑠 by d𝑡. Since 𝑠 is equal to six 𝑡 squared minus three 𝑡 plus four, we can differentiate this term by term. The velocity 𝑣 is therefore equal to 12𝑡 minus three. As the displacement was given in the standard unit of meters, the velocity will be in meters per second.

We want to know the speed of the car at 𝑡 equals three seconds. Substituting this into our equation gives us 𝑣 is equal to 12 multiplied by three minus three. The speed of the car after three seconds is therefore 33 meters per second. We recall that the momentum of any body can be calculated by multiplying its mass by its speed. This means that the momentum of the car is equal to 1,350, the mass in kilograms, multiplied by 33, the speed in meters per second. Typing this into the calculator gives us 44,550. As the mass is in kilograms and the speed in meters per second, the momentum will be in its standard units of kilogram meters per second. At 𝑡 equals three seconds, the car has a momentum of 44,550 kilogram meters per second.

We will now summarize the key points from this video. The momentum vector 𝐇 of an object is the product of its mass 𝑚 and its velocity vector 𝐯 such that 𝐇 is equal to 𝑚 multiplied by 𝐯. Momentum is typically measured in units of kilogram meters per second. As we have seen in this video, sometimes we may need to use the kinematic equations or SUVAT equations to find the velocity of an object in order to calculate its momentum. In our last example, we saw that if we are given a function for the position of an object at a time 𝑡, we can take the derivative of that function with respect to time to get a function for the velocity. This can then be used to help us calculate the momentum.