Video Transcript
In this video, we will learn how to
distinguish between velocity and speed and how to solve problems involving average
velocity and average speed.
Speed is a measure of how fast an
object moves. For example, if we’re driving in a
car, the speedometer will tell us how fast the car is going at any given time. It’s a rate at which the car covers
a certain distance. Speed is a scalar quantity. And it’s measured as a distance per
unit of time, for example, kilometers per hour. However, because it’s a scalar
quantity, the speed doesn’t give us the direction in which the object travels. If we wanted the direction, then
we’d need to use velocity because velocity is a vector quantity that gives us both
the speed and direction of travel.
For example, if a car is traveling
at 80 kilometers per hour, its speed is 80 kilometers per hour. However, if we say that the car
travels in a northeasterly direction at 80 kilometers per hour, that’s the velocity,
since we have the speed and the direction of travel of the car.
Let’s consider an object traveling
between two points, for example, this fish. There are an infinite number of
paths of different lengths that the object could take. The length of a path between two
points is the distance an object travels between the points. For this example, it’s the fish’s
distance. However, when we’re considering the
velocity of an object, we consider its displacement. The displacement measures how far
and in what direction the second point is from the first.
Let’s consider another example,
this time of a bird in flight. Let’s say that it flies 30
kilometers due east and then 40 kilometers due north. If we wanted to work out the total
distance covered by the bird, then we would add together the distances in each of
the two stages. So 𝐴 to 𝐵 is 30 kilometers, and
𝐵 to 𝐶 is 40 kilometers, giving us a distance of 70 kilometers. If instead we wanted to work out
the displacement, then that’s the directed line segment between 𝐴 and 𝐶.
We can work out the magnitude of
this line segment by recognizing that 𝐴𝐵𝐶 is a right triangle. And we could then apply the
Pythagorean theorem. This would give us the magnitude of
the displacement as 50 kilometers. In order to work out the direction,
we could take this angle 𝐵𝐴𝐶 as 𝜃 degrees. We could then note that tan of this
angle 𝜃 is 40 over 30. And so 𝜃 must be 53.130
degrees. We can then convert this into
degrees, minutes, and seconds to give us that the bird’s displacement is 50
kilometers, 53 degrees, seven minutes, and 48 seconds in the northeasterly
direction. Notice that this displacement has
both magnitude and a direction.
We can then make some notes about
speed and velocity. Speed, which is a scalar quantity,
is the rate at which an object covers distance. It’s calculated as speed is equal
to distance over time. Velocity, however, is a vector
quantity. And it’s the magnitude and
direction of an object’s change in position. It’s calculated as velocity is
equal to displacement over time. Often, however, we need to
calculate average speed. And that’s equal to the total
distance covered over the total time taken.
If we wish to calculate the average
velocity, then that’s equal to the net displacement divided by the total time. The net displacement is the
displacement measured directly from the object’s starting position to its end
position. Since displacement is a vector
quantity, then average velocity is also a vector quantity. And it can be positive or
negative. However, the magnitude, which is
scalar, is measured as a distance per unit of time. And it’s always positive. We’ll now see how we can apply
these formulas in the following examples.
An object moves north at 12 meters
per second for 10 seconds and then stops and stays motionless for 10 seconds before
moving north at 12 meters per seconds for another 10 seconds. What is the object’s average
northward velocity?
Let’s consider the three different
stages of this object’s movement. In the first stage, the object
moves north at 12 meters per second for 10 seconds. We’re then told that the object
stops and stays motionless for 10 seconds. And then, finally, it moves north
at 12 meters per second for another 10 seconds. We’re then asked to calculate the
average northward velocity. So we can recall that average
velocity is equal to the net displacement divided by the total time.
Because we have no change in the
direction, then the magnitude of this net displacement will be the same as the total
distance traveled. We can then work out the distance
in each stage. We can remember that distance is
calculated by speed times time. So in the first stage, we have a
speed of 12 and a time of 10 seconds. Multiplying those together would
give us a distance of 120 meters. In the second stage, when the
object is at rest, the speed is zero and the time is 10 seconds. So the distance traveled will be
zero meters. Finally, we have that distance in
the third stage is the same as the first stage. It’s 12 times 10, which is 120
meters.
When we add these three values
together then, we get that the total distance is 240 meters. In order to apply the formula for
average velocity, we’ll need to calculate the total time. So we have 10 seconds, 10 seconds,
and 10 seconds, which gives us a total time of 30 seconds. We can then fill these values in to
the formula, remembering that we can use the distance in this case because there’s
no change in direction here. So the magnitude of this
displacement is the same as the distance traveled. So 240 over 30 is equal to
eight. And the units here will be meters
per second. And so we can give the answer for
the average northward velocity as eight meters per second.
One other way we could’ve
approached this problem is by using a displacement–time graph. In the first stage, when the object
moved for 10 seconds, remember that we calculated that displacement as 120
meters. It then stayed still for 10
seconds. And finally, it moved another 12
meters per second for 10 seconds. In the first and third sections, we
have a positive motion. And so the graph has a positive
slope. In the middle section, this was a
rest. And so there’s zero slope. If we want to calculate the
velocity at any stage, we can find the slope or gradient of that section. If we want to find the average
velocity, then we can create a line segment between the starting coordinate and the
end coordinate.
We can remember that if we’ve got
two coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, we can calculate the slope as 𝑦
two minus 𝑦 one over 𝑥 two minus 𝑥 one. So for the two coordinates then 30,
240 and zero, zero, the slope would be calculated as 240 minus zero over 30 minus
zero. And this simplifies to 240 over 30,
which is eight meters per second. And so we have confirmed the
original answer using a displacement–time graph.
Let’s look at another example.
A person is late for an appointment
at an office that is at the other end of a long, straight road to his home. He leaves his house and runs
towards his destination for a time of 45 seconds before realizing that he has to
return home to pick up some documents that he will need for his appointment. He runs back home at the same speed
he ran at before and spends 185 seconds looking for the documents, and then he runs
towards his appointment again. This time, he runs at 5.5 meters
per second for 260 seconds and then arrives at the office.
We’re then asked three different
questions. So let’s begin with the first
one. How much time passes between the
person first leaving his house and arriving at his appointment?
It might be helpful to begin by
visualizing what happens at each stage of this person’s journey. The journey begins with running for
45 seconds towards the office. The person then realizes that
they’ve forgotten something they need, so they go home at the same speed as they
traveled before. So that means that the time will
also be 45 seconds. They then spend 185 seconds looking
for these documents but not traveling anywhere. And then, finally, he runs towards
the office at 5.5 meters per second for 260 seconds.
We’re then asked for the time that
passes between first leaving and then arriving at the appointment. So that means that we just add up
the four time periods: 45 seconds, 45 seconds, 185 seconds, and 260 seconds. And when we work that out, we get
535 seconds. And that’s the answer for the first
part of this question.
The next question asks us, what is
the distance between the person’s house and his office?
In order to find the distance, we
can use this information on the last stage of the journey, when we’re given the
speed and the time taken. We can remember that distance is
equal to speed multiplied by time. So we can fill in the values
then. The speed is 5.5, and the time is
260. It is always worthwhile making sure
that we do have the same equivalent units. In each case, the time unit is
given in seconds, so we can simply multiply these values. When we work this out, we get a
value of 1430. And the units here will be the
distance units of meters. And that’s the second part of this
question answered.
The third part of this question
asks, what is the person’s average velocity between first leaving his house and
finally arriving at his office? Give your answer to two decimal
places.
We can recall the formula that
average velocity is equal to net displacement over total time. In this problem, the net
displacement will simply be the direct distance between the man’s home and the
office. We have already calculated this
distance in the second part of the question. It’s 1430. And the total time taken in the
whole journey was 535 seconds. This gives us 2.672 and so on. And when we round that to two
decimal places, we have a value of 2.67 meters per second. So if the positive direction is
from home towards the office, then the person’s average velocity can be given as
2.67 meters per second.
It’s worth noting that if we’d been
asked for the average speed instead, we would’ve needed to know the distances in the
first two parts of the journey along with the distance in the final part of the
journey. In this case, average speed would
have been calculated by the total distance divided by the total time. However, since average velocity
uses displacement, then we have the value of 2.67 meters per second.
In the final question, we’ll see an
example where we find the average velocity when there’s a change in direction of
motion.
A man walked six kilometers in an
easterly direction for 1.2 hours. He then walked eight kilometers in
a northerly direction for two hours. Calculate the magnitude of the
average velocity of the man.
Let’s begin by thinking about this
man’s journey. To do this, we’ll need to be
familiar with our compass directions. In the first part of the journey
then, he walks six kilometers in an easterly direction. And he does that for 1.2 hours. He then walks eight kilometers in a
northerly direction, or north, for two hours. In order to calculate the average
velocity, we can recall the formula that average velocity is equal to net
displacement over total time. We can consider that the total time
in this case is relatively simple to calculate; it’s simply 1.2 plus two, which is
3.2 hours. However, we need to consider what
the net displacement is.
Well, the net displacement is the
displacement between the starting and final positions. We know that the man traveled six
kilometers easterly and eight kilometers in a northerly direction. In order to calculate this
displacement, we can observe that we have a right triangle. And so we could apply the
Pythagorean theorem. Therefore, the net displacement is
equal to the square root of six squared plus eight squared, which is the square root
of 100. And that’s 10 kilometers. And you may have already observed
that this is, of course, a three-four-five Pythagorean triple.
It’s worth noting that usually as
displacement is a vector quantity, then we would need to specify a direction of
displacement. We could do this by calculating the
angle between the horizontal and the direction of displacement. Or we could even note that the
displacement is in an approximately northeasterly direction. However, since we’re asked simply
for the magnitude of the average velocity, then we don’t need to do this here. And so we can simply fill in the
value for the net displacement into the formula to calculate the average
velocity. 10 divided by 3.2 gives us 3.125,
which we can give to one decimal place as 3.1.
The units here will be kilometers
per hour as the displacement is in kilometers and the time is in hours. And so we can give the answer that
the magnitude of the average velocity is 3.1 kilometers per hour.
We can now summarize the key points
of this video. Speed is a scalar quantity and is
the rate at which an object covers distance, where distance is the length of a path
between two points. Speed is calculated by distance
divided by time. Speed is measured as a distance per
unit of time. Velocity is a vector quantity and
specifies both the magnitude and the direction of an object’s change in
position. We calculate velocity by dividing
the displacement by the time. Velocity is given as a distance per
unit of time and has a direction.
We saw the two formulas for the
averages. The average speed is given as total
distance divided by total time. And the average velocity is given
as net displacement divided by total time. Finally, we saw that on a
displacement–time graph, the average velocity of an object’s motion is the slope or
gradient of the line between the object’s starting and finishing positions.
