The following diagram shows the
changes in the positions of two objects from time 𝑡 zero to their positions at time
𝑡 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 𝑡 zero, but at 𝑡 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
𝑡 one are shown in the diagram, but the magnitudes are not known and may be
zero. No external forces act on either
In our diagram, we can see our two
objects, the green and blue object, which are initially at time 𝑡 zero along their
path. Sometime later, at time 𝑡 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 𝑡 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 𝑡 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 𝑡 zero and at 𝑡 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 𝑡 zero and 𝑡
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 𝑡 zero and at 𝑡 one.