### Video Transcript

A car of mass 400 kilograms is moving at a constant speed of 25 meters per second when it crashes into a van moving in the opposite direction at a constant speed of 10 meters per second. The van has a mass of 700 kilograms. What is the total momentum of the car and the van? Consider the direction of the car’s motion to be positive.

Okay, so in this question, we have a car that is colliding with a van. Let’s start by drawing a quick sketch to show the information that we’ve been given. We know that we have a car moving in some direction at a constant speed. Here, we have drawn the car moving towards the right. We’re told that the speed of the car is 25 meters per second, so we can add this speed to our sketch. We’re also told that the car has a mass of 400 kilograms. So let’s label this in our sketch too.

We also have a van, which is moving in the opposite direction to the car. Since we drew the car moving towards the right, we have to draw the van moving towards the left. We’re told that the speed of the van is constant at 10 meters per second. So let’s add this speed to our sketch. We are also told that the van has a mass of 700 kilograms. So let’s label this on our sketch as well.

The question tells us that the car and the van crash into each other, and we’re asked to work out the total momentum of the car and the van. The momentum of an object is equal to the product of the object’s mass and the object’s velocity. Mathematically, an object with a mass of 𝑚 and a velocity of 𝑉 has a momentum 𝑃 given by 𝑚 multiplied by 𝑉. Let’s work out the momentum of the car and the momentum of the van. We’ll label the mass of the car as 𝑚 subscript 𝑐 so that we have 𝑚 subscript 𝑐 is equal to 400 kilograms. We will label the velocity of the car as 𝑉 subscript 𝑐.

We know that the speed of the car is 25 meters per second, and we’re told to consider the direction of the car’s motion to be positive. This means that the car’s velocity, 𝑉 subscript 𝑐, is equal to positive 25 meters per second. The momentum of the car, which we’ll call 𝑃 subscript 𝑐, is then equal to 𝑚 subscript 𝑐, so that’s the mass of the car, multiplied by 𝑉 subscript 𝑐, the velocity of the car.

Substituting in our values for 𝑚 subscript 𝑐 and 𝑉 subscript 𝑐, we have that the momentum of the car is given by 400 kilograms multiplied by 25 meters per second. Evaluating this expression, we find that 𝑃 subscript 𝑐 is equal to 10000 kilogram meters per second. This value is positive, which means it’s in the same direction as the car’s velocity.

Now let’s do the same thing for the van. We’ll label the mass of the van as 𝑚 subscript 𝑉. So we have 𝑚 subscript 𝑉 is equal to 700 kilograms. We will label the velocity of the van as 𝑉 subscript 𝑉. We know that the van has a speed of 10 meters per second, and we also know that it’s traveling in the opposite direction to the car. Since we’re taking the direction of the car’s motion to be positive, this means the direction of the van’s motion must be negative. So we have that the velocity of the van is equal to negative 10 meters per second.

Then the momentum of the van, which we’ll label 𝑃 subscript 𝑉, is equal to the van’s mass, 𝑚 subscript 𝑉, multiplied by the van’s velocity, 𝑉 subscript 𝑉. Substituting in our values for 𝑚 subscript 𝑉 and 𝑉 subscript 𝑉, we get that 𝑃 subscript 𝑉 is equal to 700 kilograms multiplied by negative 10 meters per second. Evaluating this expression gives us that the momentum of the van, 𝑃 subscript 𝑉, is equal to negative 7000 kilogram meters per second. This value is negative, which means that the van’s momentum must be in the opposite direction to the car’ s momentum.

So we now know the momentum of the car, which we’ve labeled 𝑃 subscript 𝑐. And we also know the momentum of the van, which we’ve labeled 𝑃 subscript 𝑉. The question is asking us for the total momentum of the car and the van. This total momentum is given by summing the momentum of the car and the momentum of the van.

So if we call the total momentum of the car and the van together 𝑃 subscript 𝑇, then we have that 𝑃 subscript 𝑇 is equal to 𝑃 subscript 𝑐 plus 𝑃 subscript 𝑉. Now we just need to substitute in the values that we found for 𝑃 subscript 𝑐 and 𝑃 subscript 𝑉 into this expression.

When we do this, we’ll need to take care with the signs of our values to make sure that we get the correct answer. Making the substitutions, we have that 𝑃 subscript 𝑇 is equal to 10000 kilograms meters per second, which is our value of 𝑃 subscript 𝑐, plus negative 7000 kilogram meters per second. That’s our value of 𝑃 subscript 𝑉. We can rewrite this as 10000 kilogram meters per second minus 7000 kilogram meters per second.

Evaluating this expression gives us our answer to the question that the total momentum of the car and the van is equal to 3000 kilogram meters per second. Since this value is positive, this means that the total momentum of the car and the van is in the direction of the motion of the car.