A firework is tied to a toy car to make a toy rocket car. The firework ejects exhaust from its base. The exhaust has a mass of 15 grams and moves away from the firework at a constant speed of 55 meters per second. The rocket car has a mass of 75 grams. What constant speed does the rocket car move at?
Now this is a long question with a fair amount of information in it. So let’s begin by underlining the important bits. So we’ve got a toy rocket car. Now the rocket is made from a firework. And the firework ejects exhaust from its base. This exhaust is usually in the form of gases. And we know that it has a mass of 15 grams. We also know that the exhaust moves away from the firework at a constant speed of 55 meters per second. Finally, we know that the rocket car has a mass of 75 grams. What we’re asked to do is to find the constant speed that the rocket car moves at.
Now in questions like this, there’s usually an event that takes place. In this case, the event is the rocket car moving away after their firework has ejected an exhaust. So it’s important to draw a diagram of the situation before the event and after the event. So here’s the diagram of the car before it started moving. We’ve got the rocket strapped to the top of it. And here it is afterwards. We can see the gases, the exhaust gases that the rocket releases, as the car starts moving forwards.
We can also put in some of the information we’ve been given in the question into our diagram. As soon as the car starts moving, we’ve got the exhaust fumes which have a mass 𝑚 sub 𝑒 of 15 grams and then moving to the left, in our diagram, with a velocity 𝑣 sub 𝑒 of 55 meters per second. The car and the rocket combo have a mass which we’ll call 𝑚 sub 𝑐 of 75 grams. And we need to find out what 𝑣 sub 𝑐, the velocity of the car in the rocket combo, is.
Now before the car had started moving, the exhaust fumes hadn’t been released yet from the rocket. So we could label the mass of the total system — that is of the car, the rocket, and the exhaust fumes which are to be released soon — as being 15 plus 75 grams which is 90 grams. However, the other important thing is that the whole system before it starts moving is stationary, obviously. So the velocity of the whole system before it moves is zero meters per second. And that is a reason why the mass of the total system is not irrelevant, as we will see in a second.
So now that we’ve put the most important information that we got from the question into our diagrams, we can start to think about solving the problem. To do this, we need to invoke the law of conservation of momentum. The law states that the total momentum of a system remains unchanged as long as there are no external forces acting on it. And in this scenario, that is true. We don’t have any external forces. All we’ve got is a rocket powered car doing its own thing. An example of an external force would be if we push the car with our own hands. But that’s not happening. The car is being left to its own devices. This means that there are no external forces acting on the system. And hence, the law of conservation of momentum must apply here.
So what does the law of conservation of momentum actually mean? Well, it says that the total momentum of a system remains unchanged if there are no external forces acting, as we’ve already seen is the case. That means that the total momentum of the system before anything has happened must be exactly the same as after anything has happened. In this case, the thing that’s happening is that the rocket releases its exhaust fumes. And the car starts to move. So the total momentum of a system after the car starts to move must be the same as it was before the car started moving.
Now we also need to remember how we calculate the momentum of an object or a system. The momentum of an object is found by multiplying its mass by its velocity. It’s as simple as that. And so we need to do this for the system before the car starts moving and after it starts moving. Before we do that then, let’s summarize the important bits that we’ve just learned in a little box on the side. So here are the two points. The first being momentum is equal to mass times velocity. That’s just the definition of momentum. And the second being momentum before is equal to the momentum after. That’s basically the conservation of momentum which we’ll apply to this scenario.
Now to find the momentum before the car started moving, we need to multiply the mass of the whole thing — that’s the car, the rocket and exhaust fumes — by its velocity. That’s 𝑚 sub total multiplied by 𝑣 sub total. But since 𝑣 sub total is zero meters per second, the total momentum before is zero. And of course, let’s not forget the units which in this case are grams multiplied by meters per second. And here’s the important bit. It is in grams because the mass has been given to us in grams. Of course, the standard unit of mass is actually the kilogram. But that doesn’t matter as long as we’re consistent with the units that we use. So we know that the momentum before the car starts moving is zero grams meters per second. Which means that the momentum after the car starts moving must also be zero grams meters per second.
So let’s go about finding that momentum after the car starts moving. Well, we’ve got the momentum of the moving car bit which includes the car and the rocket. And we can do that by multiplying the mass of the car rocket part, 𝑚 sub 𝑐, by its velocity, 𝑣 sub 𝑐, which is what we need to find. But of course, that’s not the total momentum of the system because we still haven’t considered the momentum of the exhaust gases. So we need to add that momentum to the momentum of the car rocket bit in order to get the total momentum of the system. And again, the momentum of the exhaust gases is found by simply multiplying the mass of the exhaust gases, 𝑚 sub 𝑒, by the velocity, 𝑣 sub 𝑒. And that is the expression for the total momentum of the system after the car started moving.
So let’s substitute our values in and see where that takes us. We know 𝑚 sub 𝑐 is 75 grams. And we need to multiply that by 𝑣 sub 𝑐 which we’re trying to find. And then we could substitute in the values for 𝑚 sub 𝑒 which is 15 grams and 𝑣 sub 𝑒 which is negative 55 meters per second. That bit is really important. The reason that the velocity of the exhaust gases is negative is because they’re moving in the opposite direction to the car. Now it doesn’t matter which direction you say is positive and which direction you say is negative as long as you’re consistent throughout the calculation. In this case, we’ve arbitrarily chosen that anything moving towards the right has a positive velocity. And anything moving to the left has a negative velocity. This is sort of makes intuitive sense because we want the direction that the car is travelling in to be positive. But like we said earlier, it doesn’t matter. All that matters is that we’re consistent. Once we say one direction is positive, then that direction has to be positive and the opposite direction has to be negative. So going back to our expression for the total momentum of the system after the car started moving, we get 75 grams times 𝑣 𝑐 plus negative 825 grams meters per second.
So all that remains is to equate the two expressions in the orange boxes. Let’s do that now. Now here we’ve said the total momentums before and after to be equal to each other. And all that remains is to rearrange the equation a little bit in order to find 𝑣 sub 𝑐. We can add 825 gram meters per second to both sides of the equation which gives us this. We’ve now got 825 grams meters per second on the left-hand side is equal to 75 grams multiplied by 𝑣 sub 𝑐. Dividing both sides of the equation by 75 grams gives us 11 meters per second is equal to 𝑣 sub 𝑐.
So our final answer is that the rocket car moves at a constant speed of 11 meters per second.