In this explainer, we will learn how to use the formula for momentum, , to calculate the momentum of objects and calculate changes in an objectβs momentum.
The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as where is the momentum, is the mass, and is the velocity.
Taking the mass of an object in kilograms and the velocity of the object in metres per second, the unit of momentum is kilogram-metres per second (kgβ m/s).
Momentum is a useful quantity when determining the magnitudes of forces that act on objects and the time intervals in which these forces act. These uses of momentum are not part of this explainer, however, which only looks at how to determine the momentum of an object or the total momentum of multiple objects.
Let us look at an example of determining the momentum of an object.
Example 1: Determining the Momentum of an Object
What is the momentum of an object that has a mass of 2 kg and moves at a constant velocity of 3 m/s?
Answer
The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as where is the momentum, is the mass, and is the velocity.
We have then that the momentum of the object is
Velocity is a vector quantity, so it has a direction as well as a magnitude. As momentum is the product of mass and velocity, it is also a vector quantity, and so it also has a direction.
In the preceding example, the direction of the velocity was not stated, and so all that can be said of the direction of the momentum is that it is the same direction as that of the velocity of the object. The direction of the velocity of the object does not affect the magnitude of the momentum of the object, so the direction of the velocity is left undefined by the question.
For any single object, its momentum can be defined as being in the direction of the velocity of the object.
Let us look at another example of determining the momentum of an object.
Example 2: Determining the Momentum of an Object
A cat has a mass of 3 kg. The cat moves 4 m in a straight line in a time of 2 s. What is the momentum of the cat?
Answer
The momentum of an object is the product of the mass and the velocity of the object. The momentum of the cat, , is given by where the mass of the cat, , is 3 kg and the velocity of the cat, , is not given.
The velocity of the cat can be determined from the fact that the cat moves 4 metres in a straight line in a time of 2 seconds. The velocity of the cat, therefore, has a magnitude given by
The 2 m/s velocity of the cat may have been constant, but it is just as valid to assume that the speed of the cat varied over the 4 metres that it traveled and that 2 m/s was only the average velocity of the cat. There is no way to determine a value of the velocity of the cat other than its average velocity, so we will take the velocity of the cat to mean its average velocity.
Knowing the average velocity of the cat, we can determine the average momentum of the cat.
We have then that the magnitude of the average momentum of the cat is and the average momentum is in the same direction as the velocity, which is along the straight-line path traveled by the cat.
The question does not specify that average momentum is required, but the only value of momentum that is possible to determine is the average momentum. We can, therefore, reasonably assume that the average momentum of the cat is equal to its momentum.
So far, we have defined the momentum of an object to be in the direction of the velocity of the object and only looked at objects that have constant velocity. The velocity of an object can change, and a change in the velocity of an object changes the momentum of an object.
Let us look at an example involving a change in the momentum of an object.
Example 3: Determining the Change in the Momentum of an Object
A child of mass 30 kg slides down a slide. Near the top of the slide, the childβs velocity is 0.65 m/s, while near the bottom of the slide, the childβs velocity is 1.35 m/s. How much does the childβs momentum change between these two positions?
Answer
The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as
The mass of the child is a constant 30 kg, but the velocity of the child changes. We can then express the change in the momentum of the child as follows:
The change in the magnitude of the velocity of the child is the value of their final velocity minus the value of their initial velocity. We can express this as
We can substitute the value of into the formula and this will give us which is the change in the magnitude of the momentum of the child. The direction of the momentum of the child before and after the change is the same as the direction of the velocity of the child: downward along the slide.
Velocity is a vector quantity, so the change in the velocity of an object when the object changes direction needs to use the rules of vector addition.
For motion along a line, velocity in one direction along the line is defined as positive, and motion in the opposite direction is defined as negative.
Suppose then that an object with a mass of 10 kg is moving left to right at a speed of 2 m/s. The object then changes direction and moves right to left at a speed of 1 m/s.
We can define positive velocity as when the object is moving from left to right. This means that when the object is moving from left to right, the momentum of the object is given by
When the object is moving from right to left, the velocity of the object is negative, so the momentum of the object is given by
The change in the momentum of the object is given by the final momentum of the object minus the initial momentum of the object. This is expressed by
The change in momentum is negative as the change is in the direction opposite to the positive direction for velocity.
Let us now look at an example of the change in momentum of an object that reverses direction.
Example 4: Determining the Change in the Momentum of an Object That Reverses Direction
A tennis ball with a mass of 60 g is thrown at a wall, hitting it while moving at a speed of 15 m/s. The ball bounces back from the wall with a speed of 10 m/s. What is the magnitude of the tennis ballβs net momentum change due to the collision?
Answer
The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as
The mass of the ball is a constant 60 grams, but the velocity of the ball changes.
The SI unit of momentum is kilogram-metres per second (kgβ m/s), so we convert the mass of the ball to a mass in kilograms. The mass of the ball is then 0.060 kg.
We can then express the change in the momentum of the ball as follows:
The change in the magnitude of the velocity of the ball is the value of its final velocity minus the value of its initial velocity.
We can define the direction of the velocity of the ball before it rebounds as the positive direction. This means that the initial velocity is 15 m/s.
After the ball rebounds, it has a velocity in the negative direction, so it has a velocity of m/s.
We can express the change in the velocity of the ball as
We can substitute the value of into the formula and this will give us which is the change in the momentum of the ball in the direction of the initial velocity of the ball.
The question asks for the magnitude of the change in the momentum of the ball. The magnitude is the positive value of the change in the momentum, so the magnitude of the change in the momentum of the ball is 1.5 kgβ m/s.
So far, we have looked at only the momentum and change in momentum of a single object. For multiple objects, the total momentum of the objects is the sum of the momentum of the objects.
Suppose we consider the total momentum of two objects moving in the same direction, which we define as the positive direction. One of these objects has a momentum of 55 kgβ m/s and the other has a momentum of 20 kgβ m/s. The total momentum of these objects is given by
Suppose instead that both objects are moving in the negative direction. The total momentum of these objects is now given by
We see from this that if the objects move in the same direction, the magnitude of the total momentum is the sum of the magnitudes of the momenta of the objects.
Now, suppose that the object with the lesser momentum was moving in the negative direction. The total momentum of these objects is now given by
Finally, suppose that the object with the greater momentum was moving in the negative direction. The total momentum of these objects is now given by
We see from this that if the objects move in opposite directions, the magnitude of the total momentum is the difference between the magnitudes of the momenta of the objects.
Let us now look at an example in which the total momentum of multiple objects is determined.
Example 5: Determining the Momentum of a System of Objects
A car of mass 400 kg is moving at a constant speed of 25 m/s when it crashes into a van moving in the opposite direction at a constant speed of 10 m/s. The van has a mass of 700 kg. What is the total momentum of the car and the van? Consider the direction of the carβs motion to be positive.
Answer
The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as
The car is traveling in the positive direction, so it has a momentum of
The van is traveling in the negative direction, so it has a momentum of
The total momentum is given by summing the momenta of the car and the truck, as shown in the following figure.
The black arrow represents the total momentum, .
We see that
The car and the truck move in opposite directions, so the magnitude of their total momentum is the difference between the magnitudes of their momenta. The car has positive momentum with a greater magnitude than the magnitude of the negative momentum of the truck, so the total momentum is positive.
Let us now summarize what has been learned in these examples.
Key Points
- The momentum of an object is the product of the mass and the velocity of the object. This relationship can be expressed as where is the momentum, is the mass, and is the velocity.
- The change in the momentum of an object of fixed mass is the product of its mass times its change in velocity. This relationship can be expressed as where is the change in momentum, is the mass, and is the change in velocity.
- Momentum is a vector quantity; hence, it can be positive or negative valued for velocity along a line.
- The total momentum of multiple objects equals the sum of their momenta.
