Question Video: Analyzing Elastic Collisions Physics • 9th Grade

The following diagram shows the changes in the positions of two objects from time 𝑑₀ to their positions at time 𝑑₁. The objects have the same mass as each other. The arrows above the objects indicate the direction of their velocities, but the magnitudes of the velocities are not known; they may or may not be equal to each other. A point 𝑃 is a distance 𝐷 from both of the objects at 𝑑₀; but at 𝑑₁, the objects are at distances 𝑑₁ and 𝑑₂ from 𝑃, where 𝑑₁ < 𝑑₂. The directions of the velocities at 𝑑₁ are shown in the diagram, but the magnitudes are not known and may be zero. No external forces act on either object.

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Video Transcript

The following diagram shows the changes in the positions of two objects from time 𝑑 zero to their positions at time 𝑑 one. The objects have the same mass as each other. The arrows above the objects indicate the direction of their velocities, but the magnitudes of the velocities are not known; they may or may not be equal to each other. A point 𝑃 is a distance 𝐷 from both of the objects at 𝑑 zero, but at 𝑑 one, the objects are at distances 𝑑 one and 𝑑 two from 𝑃, where 𝑑 one is less than 𝑑 two. The directions of the velocities at 𝑑 one are shown in the diagram, but the magnitudes are not known and may be zero. No external forces act on either object.

In our diagram, we can see our two objects, the green and blue object, which are initially at time 𝑑 zero along their path. Sometime later, at time 𝑑 one, we can see our same objects, now at new positions along their path. We are told that the objects have the same mass, but that their velocities are unknown. The arrows above the objects indicate the direction that the objects are moving but not the magnitude of their velocities. We’re gonna make room on our screen for our questions.

Which of the following is the least distance from the point 𝑃 to the point at which the objects could have come into contact with each other? (A) 𝑑 two minus 𝑑 one, (B) 𝑑 two, (C) zero, (D) 𝑑 one, (E) 𝑑 one minus 𝑑 two.

The question asked us to find the least distance from a comparison point 𝑃 right here on our diagram. We can see from the diagram that both the green ball and the blue ball are moving towards each other. We have three options for the relative motion of our objects. The blue one could be moving faster than the green, the blue could be moving slower than the green, or the blue and the green could be moving at the same speed. Because point 𝑃 is directly in the middle between blue and green, they would reach point 𝑃 at the same time if they were going at the same speed. As this is one of our options of motion, we can say this is a possibility.

If the blue object and the green object are moving at the same speed towards each other before the collision, then the distance from point 𝑃, at which the objects could have come in contact with each other, would be (C) zero.

Which of the following is the greatest distance from the point 𝑃 to the point in which the objects could have come into contact with each other? (A) 𝑑 two, (B) 𝑑 two minus 𝑑 one, (C) 𝑑 one, (D) 𝑑 one minus 𝑑 two, (E) zero.

Since the problem wants to know the greatest distance from point 𝑃, let’s start with the extreme of our answer choices. The greatest distance from point 𝑃 of our answer choices would be 𝑑 two, remembering that in the original problem, it told us that 𝑑 one was less than 𝑑 two. Looking at the motion to the objects at 𝑑 one, this must occur after the collision as we can see that that motion arrows are in the opposite directions to what they were at 𝑑 zero.

What scenario would allow our greatest distance from the diagram 𝑑 two to be correct? 𝑑 two could be the point where the objects come in contact if the green ball after the collision comes immediately to rest and the blue object would have to move a total distance of 𝑑 one plus 𝑑 two from the collision point. In the original blurb, it told us that the objects have the same mass but that the velocities were unknown. And that even though the arrow show the direction of velocity, the magnitudes may be zero. It is plausible then that the green object stops immediately after the collision and the blue object moves through a distance of 𝑑 one plus 𝑑 two. The greatest distance from the point 𝑃 to the point at which the objects could have come in contact with each other is 𝑑 two, answer choice (A).

Which of the following quantities is the value the same at 𝑑 zero and at 𝑑 one? (A) The speed of either object, (B) the momentum of either object, (C) total momentum, (D) the kinetic energy of either object, (E) total kinetic energy.

In the original blurb, it said we do not know the speeds of either of the objects. Therefore, we can eliminate answer choice (A) as this is not necessarily have to be the same at both 𝑑 zero and 𝑑 one. And since the speeds of either object are not the same, then the kinetic energy of either object does not have to be the same. Answer choice (D) can also be eliminated. And the problem does not tell us that it’s an elastic collision, which means that the total kinetic energy does not need to be conserved, answer choice (E). This leaves us with the momentum, either the total momentum or the momentum of either object.

The problem states that there are no external forces, and during a collision, we have conservation of momentum assuming there’s no external forces. Conservation of momentum tells us that the initial momentum is equal to the final momentum. This applies to the momentum of the system not the individual objects. The total momentum, answer choice (C), would be the same at both 𝑑 zero and at 𝑑 one.

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