Video Transcript
The change in the velocity of two
objects with time is shown in the graph. Which of these statements about the
speeds and distances that the two objects traveled is correct? (A) Both their speeds and the
distances they traveled are the same. (B) Their speeds are the same, but
the distances they traveled are different. (C) The distances they traveled are
the same, but their speeds are different. (D) Both their speeds and the
distances they traveled are different.
In this question, we are given a
velocity–time graph, showing the motion of two objects, and we are asked about the
speeds and distances traveled by these two objects. Now, the graph isn’t plotting the
speed of the objects, but rather their velocity. Let’s recall that velocity is a
vector quantity. That means it has both a magnitude
and a direction.
We can see that the velocity of the
red object starts out positive and later becomes negative. That means it’s initially traveling
in the positive direction, and it slows down to a stop at this point, before
changing direction and starting to travel in the negative direction. Meanwhile, the velocity of the blue
object starts out negative and later becomes positive. Unlike velocity, speed is a scalar
quantity; it has a magnitude, but no direction. The speed of an object is equal to
the magnitude of its velocity.
Knowing this, we can draw a
speed–time graph for these two objects as follows. When the objects are moving in the
positive direction, their speed has the same value as their velocity. So, for the red object, this part
of the graph when the velocity is positive looks like this on the speed–time
graph. And, for the blue object, this part
of the graph when the velocity is positive looks like this on the speed–time
graph.
When the objects are moving in the
negative direction, their velocity is negative. Since their speed is the magnitude
of the velocity, it will have the same size as the velocity, but as a positive
value. For example, a velocity of negative
one meter per second means a speed of one meter per second.
So, for this part of the graph,
when the red object has a negative velocity, its speed will look like this. Similarly, for this part of the
graph, when the blue object has a negative velocity, its speed will look like
this. We can see then that the speed of
both objects is the same at all values of time. This means that we can reject
answer options (C) and (D), as these both claim that the two objects have different
speeds.
Now, we need to think about the
distance traveled by each object. We know that both objects travel at
the same speed at all instants in time, and they do this for the same length of
time. Let’s recall that the average speed
of an object is equal to the total distance traveled by the object divided by the
time taken to travel the distance. Multiplying both sides of this
relationship by time, then after canceling the factors of time on the right-hand
side, we have that the distance traveled is equal to the average speed multiplied by
time.
Since both objects have the same
speed at all times, they must both have the same average speed. So, we have two objects with the
same average speed moving for the same amount of time. Therefore, both objects travel the
same distance. We see then that the correct answer
is option (A). Both their speeds and the distances
they traveled are the same.
