Video Transcript
In this video, we will learn how to
determine the speed of an object that moves a distance in a time.
To get started, say that we’re
standing at a particular spot and decide to move to another spot over here. And also we decide to time
ourselves as we move. So we take a step, then another,
and another, step, step, step, step, step, step, step, step, step until finally we
arrive. We traveled a distance of 15 steps,
and we did it in a time of 15 seconds. We can say that we moved at a
certain speed. That speed equals the distance we
moved divided by the time taken to move that distance. In general, a speed is always a
distance divided by a time. Let’s look at a quick example.
Which of the following is the
correct formula for the speed of an object? (A) Speed equals distance moved
multiplied by time moved for. (B) Speed equals distance moved
divided by time moved for. (C) Speed equals time moved for
divided by distance moved.
To see which formula is correct,
let’s think about an object in motion. The faster this object moves, the
greater its speed will be. A higher speed means the object
moves a greater distance in the same amount of time. Or the object could move the same
distance but take less time. We can say the object’s speed
increases in two different cases: first, when it moves a greater distance in the
same amount of time and second when it moves the same distance in less time. Looking at our answer choices, the
only one that is true for both of these conditions is option (B) speed equals
distance moved divided by time moved for.
Let’s now return to our situation
of walking from one point to another. Remember that we moved from the
start to the end in 15 steps, and it took us 15 seconds. Let’s say that each one of those
steps covered one meter of distance. This means that after our first
step, we had traveled one meter. Then after the second step, we had
traveled a total distance of two meters, then three meters after the third step, and
so on, with the person taking 15 steps. So the total distance moved is 15
meters. But then, remember that at each one
of these steps, one more second of time had passed. When we had traveled one meter, one
second had elapsed, then two seconds at two meters, three seconds at three meters,
and so on. For every second of time passed, we
moved the same distance, one meter.
Whenever motion happens at equal
distance intervals over equal times, that motion happens at what we call uniform
speed. This means the speed is always the
same. That is, the object moving always
travels equal intervals of distance in equal times.
Let’s now consider another
example.
Fill in the blank: If an object
moves with a uniform speed, then it moves blank distances in blank times. (A) Unequal, equal; (B) unequal,
unequal; (C) equal, equal; (D) equal, unequal.
We’re talking here about an object
that moves at uniform speed. Say that this is our object, and
then it’s moving to the right. Since the object moves with uniform
speed, that means every time it moves a certain distance to the right, that moving
always takes the same amount of time. In other words, moving equal
distances requires equal amounts of time. Our completed sentence reads as
follows: If an object moves with a uniform speed, then it moves equal distances in
equal times.
So far, we’ve seen that this
sentence is true. An object’s speed equals the
distance it moves divided by the time taken. Another way to say this is using an
equation. The equation has to do with speed,
distance, and time. This sentence tells us that speed
is equal to distance divided by time. This equation says the same thing
that the sentence says, and we can even write it in a shorter way. We can represent the quantity speed
by the letter 𝑣. Distance we’ll represent with a 𝑑
and time with a 𝑡. Then, we can rewrite this
equation. Using symbols, we can say that
speed 𝑣 equals distance 𝑑 divided by time 𝑡. This equation in symbols says the
same thing as the equation in words. And that says the same thing as
this sentence.
Let’s now get some practice using
this equation for speed.
A bike moves uniformly. The position of the bike at two
different times is shown. What is the speed of the bike?
In this diagram, we see the bicycle
at zero seconds and two seconds. Over this time, it’s traveled a
distance of 10 meters. We want to solve for the speed of
the bike. We’ll call the speed 𝑣. In general, speed is given by this
equation. The speed of an object 𝑣 is equal
to distance it travels, 𝑑, divided by the time taken to travel that distance
𝑡. With our bicycle, we know the
distance it travels. That’s 10 meters, so 𝑑 is 10
meters. Then, what about the time 𝑡? We see the bike’s position at zero
seconds and two seconds. That means the total time is two
seconds.
Now that we know 𝑑 and 𝑡, we can
use our equation to solve for speed. 𝑣 is equal to 10 meters divided by
two seconds. In other words, it’s 10 divided by
two meters per second. 10 divided by two is five. So the speed of the bike is five
meters per second. The magnitude of this speed is
five, and the units are meters per second.
Let’s now look at another
example.
An aircraft moves at a speed of 630
kilometers per hour for eight hours. How far does the aircraft move?
Let’s say that this is our
aircraft. We know how fast it’s traveling,
630 kilometers per hour. And we know for how long it
travels, eight hours. We want to know how far it moves,
its total distance. We can think of it like this. If the aircraft moves this far in
one hour, then in eight hours, it will travel this distance. We want to know this distance that
we’ll call 𝑑. It’s related to the aircraft’s
speed and the time passed. In general, the speed, 𝑣, of an
object equals the distance it travels, 𝑑, divided by the time, 𝑡. In our case, the speed 𝑣 is 630
kilometers per hour. The time 𝑡 is eight hours.
But look at this. This equation lets us solve for
speed. However, it’s distance that we want
to know. What we’ll do is rearrange this
equation. We multiply both sides by the time
𝑡. Since we’re multiplying each side
by the same value, we’re not changing the equation. By doing this though, the 𝑡 in the
numerator on the right and in the denominator cancel out. We end up with this equation. Distance 𝑑 equals time 𝑡
multiplied by speed 𝑣. And we know 𝑡 and 𝑣, so we can
substitute those values in. The distance this aircraft moves is
eight hours times 630 kilometers per hour.
And notice something here with the
units. In our time of eight hours, we have
a unit of hours in the numerator. And this time multiplies a speed
with a unit of hours in the denominator. That means when we calculate 𝑑,
these units of hours will cancel out. Our final answer will have units of
kilometers. When we multiply eight by 630, the
result is 5040. And as we said, this has units of
kilometers. This distance is how far the
aircraft moves.
Let’s look at one more example.
A train travels 160 meters at a
uniform speed of eight meters per second. The position of the train at two
different times is shown. How much time does the train take
to travel the distance between the positions?
In this diagram, we see the train
moving a distance of 160 meters while traveling at a uniform speed of eight meters
per second. We want to know how long it takes
for the train to travel this distance. In general, the speed of an object,
𝑣, is equal to the distance it moves, 𝑑, divided by the time taken to travel that
distance, 𝑡.
In our example, we know the train’s
speed and distance traveled. Its speed is eight meters per
second, and the distance is 160 meters. What we want to solve for is time
𝑡. To begin doing that, let’s
rearrange this equation. First, we’ll multiply both sides by
the time 𝑡. On the right side of the equation,
𝑡 now appears in the numerator and the denominator. That means it cancels out. And that gives us this
equation. And our next step will be to divide
both sides by 𝑣. We do this because we want the time
𝑡 all by itself on one side. Notice what happens on the
left. The 𝑣 in the numerator and the
denominator cancel one another out.
Now, we have an equation where 𝑡
is the subject. It’s equal to distance divided by
speed. We know the distance and speed for
our train, so we can plug those values in. The time taken for the train equals
160 meters divided by eight meters per second. Eight divided into 160 equals
20. And in the units of meters divided
by meters per second, the units of meters will cancel out. We’re left only with units of
seconds. The time the train takes to travel
160 meters at a speed of eight meters per second is 20 seconds.
Let’s finish up this lesson by
reminding ourselves of a few key points. In this lesson, we saw that speed
equals the distance an object travels divided by the time taken to travel that
distance. Written as an equation, 𝑣 equals
𝑑 divided by 𝑡, where 𝑣 is speed, 𝑑 is distance, and 𝑡 is time. An object moving with uniform speed
travels equal distances in equal times. Lastly, we saw that the equation 𝑣
equals 𝑑 divided by 𝑡 means that 𝑑 equals 𝑣 times 𝑡 and 𝑡 equals 𝑑 divided by
𝑣.
