Video Transcript
The diagram shows two freely moving particles in a container. Both particles have the same mass. The positions of the particles shown are the highest points above the base of the container that the particles reach. Neither particle has any horizontal motion. What is the ratio of the kinetic energy of the higher particle to that of the lower particle when both particles are at the base of the container?
Okay, so in this question, we can first of all see that we’ve got a container containing two particles. We’ll call these particle one and particle two. We’ve been told that these particles are freely moving. So nothing is restricting their motion in any way. Additionally, we’ve been told that both particles have the same mass. Now, the position shown, the positions of the particles one and two in this diagram are the highest points above the base of the container that the particles reach. So practical one reaches the height of ℎ one as its highest point above the base of the container. And particle two reaches a maximum height ℎ two.
But then, if these heights ℎ one and ℎ two are the maximum heights of particle one and particle two, respectively. Then from this we can gather that the velocity of particle one and particle two in the vertical direction at these points must be zero. So what do we mean by this? Well, let’s consider particle one first of all. At the point where it’s at a height of ℎ one, the particle cannot be moving in an upward direction. Because otherwise that wouldn’t be the maximum height of particle one. It would continue to move upwards to a higher point. And similarly, it could not be moving in a downward direction with any velocity. Because otherwise it means that it would’ve had a velocity of zero at some higher point. And it’s now falling downward.
Therefore, if ℎ one is the highest point of particle one, then the upward or downward velocity must be zero. Granted the velocity of the object will increase in the downward direction due to the pull of gravity. But specifically at height ℎ one, at the maximum height, the velocity of the object is zero. And the same is true for particle two.
Now, as well as this, we’ve been told that neither particle has any horizontal motion Therefore, both particles will only ever move up or down. And the importance of this is that at height ℎ one for particle one, the velocity of particle one is zero. Not just in the up and down direction but in any direction. At height ℎ one, particle one is stationery. And similarly for particle two, at height ℎ two, that particle is stationary.
But then, if at those heights these particles are stationary. Then that means that neither of these particles has any kinetic energy at these points. The reason we can say this is because we can recall that kinetic energy is defined as half multiplied by the mass of an object multiplied by the velocity of the object squared. And at point ℎ one, we’ve said that the velocity of particle one is zero. And, hence, the kinetic energy of particle one at height ℎ one is also zero. This means that at its maximum height, the only kind of energy particle one has is gravitational potential energy.
And we can recall that, generally speaking, the gravitational potential energy is defined as the mass of an object multiplied by the gravitational field strength of earth multiplied by the height above ground that the object is at that point. And so, we can say that for particle one at height ℎ one, it has a gravitational potential energy of 𝑚, which is the mass of particle one, multiplied by 𝑔, the gravitational field strength of earth, multiplied by height ℎ one. Because we’re saying that the bottom of the container is ground level. And so particle one is ℎ one above this ground level.
And crucially, 𝑚𝑔ℎ one, the gravitational potential energy of particle one, is all the energy that particle one has. Remember, we saw already that at this point there is no kinetic energy of the particle. And so, the total energy of particle one is 𝑚𝑔ℎ one which is the gravitational potential energy. And similarly for particle two at ℎ two, we can say that it has a gravitational potential energy 𝑚, which is the mass of particle two. And, remember, the mass of particle two is the same as the mass of particle one. And so, we multiply the mass of particle two by 𝑔 and by ℎ two. And once again, very crucially, this is all the energy that particle two has. It doesn’t have any kinetic energy. And so, the total energy of the particle at height ℎ two is simply 𝑚𝑔ℎ two, the gravitational potential energy.
Now, coming back to our question, we’ve been asked to find the ratio of the kinetic energy of the higher particle, particle two, to that of the lower particle, particle one, when both particles are at the base of the container. In other words, when particle two falls down to the base of the container, we need to work out its kinetic energy at this point here. And, similarly, we need to work out the kinetic energy of particle one at this point here. And find the ratio of the kinetic energy of particle two at this point to the kinetic energy of particle one at this point.
To work this out, what we need to realize is that as the particles fall, they will lose gravitational potential energy. Because they’re getting closer to the ground. So they’re losing their height. And, hence, the value of the gravitational potential energy will decrease. But then, where does all of this gravitational potential energy go? Remember, energy cannot be created or destroyed. So it must be transferred somewhere. And the answer to that is that it becomes the kinetic energy of the particle. Because as the particle falls, its velocity increases. And this is true for both particles. As they both fall, their velocities increase.
So even though they had zero velocity at their highest points, the velocity at the base is not going to be zero. It’s going to be much higher than zero. And once both of the particles reach the bottom of the container, all of their gravitational potential energy will have been converted to kinetic energy. Because at the base of the container, their gravitational potential energy is zero. Because the height at the base of the container is zero. And that is why we can say that all of the gravitational potential energy they had initially has been converted to kinetic energy. And so, we can say that the kinetic energy of particle one at the base of the container will of course be given by half multiplied by the mass of the particle multiplied by the velocity of the particle at the base squared.
But, we can also say that this must be equal to the initial gravitational potential energy that the particle had. Which was the gravitational potential energy at the top of the trajectory, 𝑚𝑔ℎ one. Now, it might look very strange to say that kinetic energy is equal to 𝑚𝑔ℎ. But that’s not what we’re doing here. We’re equating the kinetic energy at the bottom to the gravitational potential energy at the top. And so, there’s no problem with our saying this here. And similarly for particle two, we can say that the kinetic energy of particle two at the base of the container must be equal to the gravitational potential energy of that same particle at the top of its trajectory, 𝑚𝑔ℎ two.
And, at this point, we found expressions for the kinetic energy of particle one at the base of the container and the kinetic energy of particle two at the base of the container. And happily for us, the question is asking us to find the ratio of the kinetic energy of particle two, the higher particle, to that of the lower particle, particle one. In other words, what we’re being asked to do is to find the ratio of the kinetic energy of particle two at the base of the container to the kinetic energy of particle one at the base of the container. And we can substitute the expressions that we found above. We know that this is equal to 𝑚𝑔ℎ two divided by 𝑚𝑔ℎ one. And this is where it’s important that we account for the fact that the masses of the two particles are the same.
We’ve been told in the question that both particles have the same mass. Because this allows us to cancel the mass in the numerator and the denominator. They’re the same value. So we’re dividing the mass by the mass, and hence it cancels. And the same is true for the gravitational field strength of the earth. We’ve got the value of 𝑔 in the numerator and the denominator, and so it must cancel. And so, what that leaves us with is simply ℎ two divided by ℎ one. The maximum height of particle two divided by the maximum height of particle one. And that is the ratio of their kinetic energies at the bottom of the container. Hence, at this point, we’ve found the answer to our question.
