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# Velocity vs. time graphs

Let's learn how to calculate accelerations from velocity-time graphs. We will also get some intuition behind why slope or steepness of a velocity-time graph represents acceleration. Created by Mahesh Shenoy.

## Want to join the conversation?

- why do we write 0 twice in the x axis?(3 votes)
- Actually 1 is for x and other for y .it is okay if you write once only(4 votes)

- What will be the shape of the graph if the velocity of the object is constant ?(3 votes)
- Linear, Straight Line..(1 vote)

- Is there a jerk-time graph (jerk in means of change in acceleration over time?(2 votes)
- yes, there is one(1 vote)

- What will be the shape of the graph if the velocity of the object is constant ?(1 vote)
- a straight line(2 votes)

- But in 1 dimensional motion, if the both objects are travelling in same direction, the would catch up one day right?(1 vote)
- Depends if their acceleration changes, the graph only shows the velocity and time of that extent, no one knows what happens after(1 vote)

- How would a velocity vs time graph work with a parabola?(1 vote)
- if it is a parabola, then instantaneous acceleration will change depending on type of parabola along with overall length of the chord(1 vote)

- A car travelling due north at 60 km/h increased its velocity to 80 km/h in 20 seconds, draw acceleration - time graph.(1 vote)
- 1:19he said downwards meaning time can go negative. but this is not possible(0 votes)
- He's not talking about time, he's talking about velocity. He's saying that when velocity is negative, it is in the direction opposite to the direction that is being considered as positive.

For example, if we consider moving upwards as positive, then the velocity of -4m/s would be a velocity of 4m/s in the downwards direction.

Hope you understood.(6 votes)

## Video transcript

- A cheetah is running somewhere and you would love to see it. But since you like physics so much, instead of showing you the clip, I give you it's a velocity-time graph. Whew, So let's see if we
can analyze this graph and visualize what our cheetah is doing. So, what's a velocity-time graph? It's a graph that tells us
the velocity of any object, like our cheetah, at
every moments in time. I like to think of it as
watching a speedometer of a car. It is just like that. The speedometer tells us what
the speed of our vehicle is at every moments in time. The only difference is, here, it not only gives us the speed, not only tells us how fast it's going, but it also tells us in what direction that object is moving. That's what velocity is. It's speed plus direction. How does this graph give us a direction? Well, if the graph is
positive, like right now, like you can see over
here about the time axis, then we can say the object is moving in one particular direction, let's say to the right, or
maybe upwards as an example. And if the graph were to
come below the t-axis, then we would say the
velocity is negative. Meaning it's in the opposite direction. It could mean to the left or it could mean downwards, anything. And so the graph gives us both
the speed and the direction. That's why it's called
velocity-time graph. But let's not worry too
much about the direction, we'll only concentrate on the speed part. So what is the graph saying? Well, the graph is saying
that at time, t equal to zero, our cheetah has zero speed. But then as the time ticks,
look at what's happening. It's speed is increasing. Can you see that, one second
it's speed is increased. It's speed increases. So it's speed is continuously increasing and at the end of five seconds, it has reached 20 meters per second. So if we had a speedometer
attached to our cheetah, then it would look somewhat like this. At time, t equal to zero
when we just start our timer it has zero speed, but then it quickly accelerates. Its speed increases and at
the end of five seconds, it reaches a spirit of
20 meters per second. And we don't know what happens after that because the graph ends over there. And now I'm guess we can
pretty much visualize what our cheetah is doing. So, let's say here is our
one of a kind pink cheetah. We know that at time, t equal
to zero it's speed is zero. So if we attach this speedometer to it, then as time ticks, it gets
faster and faster and faster, and at the end of five seconds, it hits 20 meters per second. Now, one important thing
to think about is that just like when you're
looking at the speedometer of a vehicle, it only tells us how fast it's going, but it doesn't tell us
where that object is, or where that vehicle is, right? Similarly, the velocity-time graph, also tells us only the
speed, how fast it's going, and it tells us in what
direction is going. But it doesn't tell us
anything about its location, its position. So I have no idea where this
cheetah was to begin with. It could have been at zero, but it could have been a hundred meters or maybe it would have been
that minus a hundred meters. There is no information
about its position, but only about its speed and direction. Okay, now that we know our cheetah is getting faster and faster, it means it's accelerating, right? Whenever objects change their speed, whether they increase it or decrease it, we say that object is accelerating. So, can we calculate this acceleration by looking at this graph? Well, let's see. How do we calculate
acceleration in general. We define acceleration, a,
as the change in velocity divided by time. Change in velocity is always calculated as the final velocity, v
minus initial velocity, u. So this is the change in
velocity divided by time. So let's see if we can look at
our graph and calculate this. In fact, you know what? See if you can try this on yourself first. Go ahead, pause the video
and see if you can do this. If you can calculate the acceleration by looking at the graph. All right, let's see. We know at t equal to zero,
it's velocity is zero. Let's call that out as our initial u. And we know at five
seconds, its velocity is 20. Let's call that as v, that's our final. And so acceleration becomes final velocity minus, it's meters per second, minus initial velocity, which is zero. Do you have the time
taken for that change? And the time taken was five seconds. And that number equals 20 divide by five. So it's five seconds. 20 divided by five, that's going to be four
meters per second per second, Which we call four meters
per second squared. That is the acceleration of our cheetah. What does it mean? It means our cheetah is gaining a speed of four meters per second every second. Every second it's increasing its speed by four meters per second. That's the acceleration of our cheetah. Okay, now let's say there's a deer that is chasing this cheetah. Whew! Things are getting very interesting. And it's velocity-time graph is over here. We see that even this
deer is accelerating. Now, can you figure out
what is the acceleration of this deer? Is it more than the
cheetah or less than that? Again, pause the video and
see if you can do this. Okay, let's see. Again at time zero, our
deer's also be zero. We can call that as initial. And then at the end of two
seconds, it reaches 20. So we'll call that as final. So the acceleration for our
deer is going to be v minus u by the same formula. 20 minus zero divided by two seconds, because it reaches 20 in only two seconds. And so that would equal 20 divided by two. That is 10 meters per second per second, or 10 meters over second squared. which means the deer has
a way higher acceleration compared to the cheetah. That makes sense, right? The cheetah took five seconds
to increase its speed to 20 and our deer only took two
seconds to increase its speed. So it has a higher acceleration. And in this crazy world, because our deer is chasing the cheetah, can we say that the deer
will quickly catch up to it? Need not be. Because remember, we don't know anything about
their initial locations, initial positions. It could be totally possible
that the cheetah is miles ahead of the deer to begin with. So we can't comment on
how long it would take to catch up at all. Another most important thing
we can see from the graph is that the deer which
has a higher acceleration has a much steeper graph
compared to the cheetah. I mean, if you were to
think of these as mountains which you are climbing, then notice, climbing this
mountain is much steeper. It's much harder than
climbing this one, right? And so this means that
in a velocity-time graph, if the graph is steeper,
more acceleration. If the graph is less
steep, less acceleration. And that's why we like this graph because just at a glance,
by looking at the steepness, we can figure it out, which has more, and which
has less acceleration. And this also helps us
understand one more thing. Again, if you were to
say climb this mountain, then regardless of where you are, whether you are over here or here or here, the steepness remains
exactly the same, right? I mean it's not any harder
to climb the mountain here compared to over here. Since the steepness remains the same and the steepness represents acceleration, we can also say that the
acceleration of that cheetah is a constant. Meaning, the cheetah is gaining velocity. Its velocity is changing, but it's acceleration
does not change with time. It means every second it will
gain four meters per second of velocity, constant acceleration. Similarly, even for the deer,
it's a constant acceleration. But on the other hand, if the
graph was not a straight line, but instead say it looked like this, then it's not a constant acceleration. I mean notice in the first two seconds, it gains a lot of velocity,
very high acceleration. The next two seconds,
it gains a little bit. The acceleration has decreased. And then after four seconds notice, it's not gaining anything. The velocity remains 20. Acceleration becomes zero. Even if you look from the
steepness point of view, can you see that
initially it's very steep. So high acceleration, less
steep, low acceleration. And then the acceleration becomes zero. The graph is flat. Acceleration is zero. And so if you had a graph
that looks somewhat like this, the acceleration is not a constant, it's initially very high and then the acceleration decreases. But for a straight line, it means the acceleration
remains a constant. And to understand the difference even more let's bring back our speedometer. So, if you could look at the speedometer for this kind of motion, we would see initially
the stick moves very fast because the pickup is very
high and then it slows down. Okay, here it is. Notice, initially it was very fast, later it slows down, slows down. And then it's almost a constant. But if we again look at the speedometer for this kind of motion, well notice the pickup is uniform. Can you see that? It's uniformly picking up. And it's for that reason
this is also called uniformly accelerated motion. That means its velocity is changing, but it's acceleration is a constant. And so what did we learn in this video? We learned how to analyze
the velocity-time graphs. We saw that the steeper the
graph more is the acceleration. And we also saw it does not tell us about the position of an object. It doesn't tell us where
the object is to begin with, but it only tells us about
how fast things are moving.