Video Transcript
Two athletes run in a rectangular
field. They both start at the same
position, in a corner of the field. Athlete A runs across the field
until she reaches the opposite corner. Athlete B runs along the east-west
running edge of the field until he reaches the corner adjacent to the starting
position. This is shown below. Both athletes leave the starting
position at the same time and reach their finishing positions at the same time. Which of the following statements
is true? (A) Athlete A’s velocity in the
east-west direction is greater than athlete B’s velocity in the east-west
direction. (B) Athlete B’s velocity in the
north-south direction is greater than athlete A’s velocity in the north-south
direction. (C) Athlete A’s resultant velocity
in her direction of travel is greater than athlete B’s resultant velocity in his
direction of travel.
In this question, we are given a
figure showing a rectangular field and the motion of two athletes as they move
across it. We are told that both athletes
start in the same position in the corner of the field. Athlete A moves diagonally across
the field to the opposite corner, while athlete B moves horizontally, along the
east-west direction, to the adjacent corner. We are asked to figure out which of
these given statements are true if both athletes leave the starting position at the
same time and reach their finishing positions at the same time.
Let’s begin by drawing the vectors
representing the velocities for both of the athletes. Let’s recall what these velocity
vectors can tell us about the motion of the athletes. Recall that a vector has the
quantities of direction and magnitude. And they can be broken down into
the component of the vector in the vertical 𝑦-direction and the component of the
vector in the horizontal 𝑥-direction. The total velocity vector of each
athlete is simply their 𝑥- and 𝑦-components added together, like so.
Looking at the two velocity vectors
next to each other, we can see that their 𝑥-components must be the same, as they
travel the same distance in the east-west direction in the same amount of time. However, their 𝑦-components are
not the same. Athlete B doesn’t move in the
𝑦-direction at all, meaning that B 𝑦 must be zero. This makes the total 𝐁 velocity
vector just equal to B 𝑥.
Now that we have this information,
let’s look at the answers we’re given and determine the correct answer.
The first option states that
athlete A’s velocity in the east-west direction is greater than athlete B’s along
the same direction. We know that this isn’t true
because the velocity in the east-west direction is the same for both athletes.
The next option states that athlete
B’s velocity in the north-south direction is greater than athlete A’s velocity in
that direction. But we know that athlete B has no
velocity in the north-south direction, so this can’t be true.
Looking at the final option, it
states that the resultant velocity of athlete A in her direction of travel is
greater than the resultant velocity of athlete B in his direction of travel. Recall that the resultant velocity
is equal to the sum of the vector component in the east-west, 𝑥, direction and the
component in the north-south, 𝑦, direction. Now, we know that the east-west
component of both athletes are equal to each other. And only athlete A has a component
in the north-south direction, which means that the resultant velocity for athlete A
is greater than the resultant velocity for athlete B.
When we look at the original
diagram of their motion, we can see that this makes sense because the distance from
the starting position to the ending position is larger for athlete A. And we know that it takes both
athletes the same amount of time to reach their ending positions. So, athlete A will need to have a
greater velocity to travel a greater distance than athlete B in the same amount of
time.
Therefore, the correct answer is
option (C). Athlete A’s resultant velocity in
her direction of travel is greater than athlete B’s resultant velocity in his
direction of travel.
