Calculating Change in Velocity from Acceleration-Time Graphs - Online Tutor, Practice Problems & Exam Prep

To calculate changes in velocity from acceleration-time graphs, find the area under the curve, similar to how displacement is derived from velocity-time graphs. The area represents the change in velocity (Δv). For example, if a box starts at rest, the actual velocity at a given time can be determined by adding the initial velocity to the calculated change in velocity. Use the formula for area: for rectangles, area = base × height, and for triangles, area = ½ × base × height. This method allows for understanding motion through acceleration and velocity relationships.

Calculating Change in Velocity from Acceleration-Time Graphs

Hey, guys. So sometimes, you'll have to calculate changes in velocity, from the acceleration time graphs instead of using your equations. So that's what we're gonna check out in this video, but we're gonna see that it is exactly like how we calculated displacements from velocity time graphs. Let's check it out.

So guys, when we had velocity time graphs, the area under the curve, remember, was just the area that's enclosed between the values of the graph and the time axis. So you just make this little shape here, and then your area under the curve represents your change in the position or the displacement, Δx. Well, for an acceleration-time graph, it's the same exact principle. The area under the curve is just the area between the acceleration values and the time graph, except now the area that's enclosed within the shape here doesn't represent the displacement. It represents your change in the velocity. So it's the same idea, it's just the variable that's different here. So that's really all there is to it, guys. It's the same exact procedures. We're just gonna be calculating a bunch of areas. So let's go ahead and check out this example.

We've got an acceleration-time graph for a moving box, and we're told it's initially at rest. And so part a says we're gonna calculate the box's velocity at t=3. So we know how to do this. We just figure out t=0, t=3, and then we just have to split this up, and we have to calculate the area that's inside of this shape over here. This is not a very simple shape, and I don't know what the formula off the top of my head, but I can break it up into a rectangle or square and a triangle. And so I can calculate these areas, which should be ΔV1, I'm gonna call that, and ΔV2. And now I just have to figure out the total velocity by calculating and adding up these two things.

Now I'm also told in part a that it's not the change in velocity that I'm looking for, it's the actual velocity. So what does that mean? Well, what it's asking for is v, but I know how to calculate the velocity by first calculating the change in the velocity by looking at the area of the graph. Now remember that the change in the velocity, Δv, is just v_final−v_initial. And so, and I'm told that initially the box is at rest. So I know that this v0 is just equal to 0. So the box's velocity and the change in the velocity are the exact same thing.

So that's really all it's asking me for. So I'm gonna look for the velocity and I'm gonna call this v3, and all I have to do is just add up these 2, shapes together, ΔV1 and ΔV2. So ΔV1 is just a rectangle. It's just base times height, that's the area. Remember the base is 2 and the height is 2. So that means that's just gonna be 2 times 2, which is 4. Now the units for this are gonna be in meters per second. Remember, because it is a change in the velocity. So you have to be careful that it's meters per second.

Now ΔV2 is just triangle. So it's gonna be 12base×height. That's the formula. The base of this is 1, the height of this is 2. So it's gonna be 12×1×2. The one-half and the 2 will cancel, and we'll just get 1 meter per second. So now we just have to add these 2 sort of like smaller values and that's gonna be the total velocity at 3 seconds. So it's just gonna be 4 plus 1, and that is 5 meters per second.

That's all there is to it. So now let's figure out what the velocity is at t=5. So all we have to do is now figure out the area under the curve from 0 to 5 seconds. And so that's gonna include everything from 0 to 3, plus now from 3 to 5. So that's all we have to do is we just have to tack on this extra area that we have between this little triangle here and the graph. So I'm gonna highlight this in green. So this is gonna be the area that I have to include now. It's not a very pretty shape, so what I can do is just break it up into 2 small right triangles here. I'll call this 1, ΔV3, or sorry, ΔV4. But what you'll see is that they're symmetrical. They're the exact same pieces. So whatever I find for one value, it's gonna be the same for the other one.

Okay. So the value or the velocity at 5, I'm gonna call that v5, is just gonna be delta v. It's actually going to be, whoops. It's gonna be the velocity that I found out in part a. It's gonna be the 5 meters per second plus, now it's gonna be ΔV3 plus ΔV4. And so, I already have what this is. I know this is 5 meters per second. And now plus, I can just add up together these other two velocities here, and this is gonna be my total velocity in meters per second. So that's the plan.

So, I've got ΔV3. I know this is triangle, so I'm gonna use 12base×height. The base is 1, and the height is 2. So I'm gonna have 12×1×2, and that's again just going to give me 1. But again, remember that when you have areas that are above the time axis, those are gonna be positive changes in your velocity because your acceleration is positive. Whereas if you have negative, or sorry below the time axis, then those are gonna be negative changes in ΔV because your acceleration is negative. So your velocity is becoming more negative. So that's really all there is to it. So that means this is a negative one, and ΔV4 is the same exact thing. It's also negative one. So this is gonna be negative one and negative one, and so your total velocity at 5 seconds is 5 minus 2, which is going to be 3 meters per second, and it's gonna be positive.

So those are our two answers. Let me know if you guys have any questions. Let's get some more practice.

Sliding Block

Hey guys, let's work this one out together. We've got an acceleration time graph that's a versus t for a sliding block and we're told that the initial velocity is 3 meters per second and we're supposed to find what's the block's final velocity at t equals 5 seconds. So let's check it out. What is it that we're trying to find? Well, we're actually trying to find what the final velocity is and so this is going to be our target variable. How do we find the final velocity? Well, first, we just need to figure out what's the change in the velocity in general. The change in the velocity is just going to be my v final minus v initial. Velocity, delta v, is actually just going to be the area that is under the acceleration-time graph. And so we can rearrange this equation over here and figure this out. And so the final velocity, or v, is just going to be the initial velocity plus the change in the velocity. In other words, just the area that's underneath the curve here. And so I already have the initial velocity that I'm given. It's 3 meters per second. And so all I need to do is if I try to figure out the final velocity, I just need to figure out what's the change in v. So in other words, what's the area that's underneath the curve? So let's go ahead and figure that out. Basically, I know that this is from t equals 0. Okay. So that's my initial velocity and I have to figure out the final velocity, t equals 5. So that means that basically I'm going to be adding up all of the area that is between these two points on the curve. And the area that that corresponds to underneath the graph is just everything that's going to be included between the values and the time axis. So it's going to be all the stuff that's in green over here. Now again, this is weird. It's kind of like this triangle looking thing, but it's not really a nice shape that I need. So what I can do is break it up into more manageable shapes. So what I can do is I can draw a line down here, and this will be one triangle. And then I can draw a line over here, and that'll be a rectangle and another triangle. So this should work. So now I'm going to have 3 shapes over here. And so remember, I'm going to have to find out what delta V is, so let's write out an equation for this. My delta v is just going to be if I have these 3 smaller shapes, it's just going to be adding up all of the areas that are underneath those, those shapes. So I'm going to call this one delta v one, I'll call this guy delta v two, and I'll call this guy delta v 3. And you just have to use the right formulas for each one of them. So let's just get to it. The total change in the velocity is just going to be once I add up all of the changes in velocity. Delta v 1, delta v 2, delta v 3. Okay. So let's get to it. So delta v one is going to be a triangle, which means the area we're going to use is math>1/2 base times height. So now, I just have to figure out the base and the height. So I've got 1, 2, 3. The base of the triangle is going to be from 0 to 3, so that's 3. The height is going to be from 2 all the way up to 6. So that means that the height of this triangle is 4. So I've got 1/2 of 3 times 4, and so that's going to be 12. 1/2 of 12 is 6. And remember, these are going to be in meters per second. So delta v 2 is just going to be a rectangle. Rectangle, so you just use base times height for this, and the base of this triangle, or sorry, this rectangle's going to be 3, the height of this is going to be 2, and so_we just have 3 times 2, and that's also 6 meters per second. And then, now finally, let me just rewrite this actually, so that's delta v 3. So I've got 1/2 of base times height, and so, the base of this triangle is going to be 2. The height of this triangle is actually going to be 6, because we're going from 0 all the way up to 6 here. So just make sure you have your bases and your heights, sort of worked out. So we've got 1/2 of base, which is 2, height is 6. 1/2 and the 2 will cancel, and then we'll just end up with 6 meters per second. So this is kind of interesting. Notice how we get 6 meters per second for all the shapes. 6, 6 and 6, even though they all kinda look different. Obviously, all your numbers won't work out to always 6. You have to, you know, do the do the math for each one of them. But it's kinda interesting how all these different looking shapes, we ended up ended up getting the same area. So we actually have all of these three things here, which means we can calculate the total change in the velocity. Delta v, it's just 6 plus 6 plus 6, which equals 18 meters per second. So now, are we done? Is this our answer? Well, no. Remember, this represents the change in the velocity, not the final velocity. So we actually have to plug this back into this equation over here to figure out the final_velocity. So let's go ahead and do that. Our final velocity is going to be our initial velocity of 3 meters per second. So it's going to be 3 plus now the change, and now_pain 18. So_blog_ final velocity of 21 meters per second, and this is our final answer. Alright, guys. Let me know if you have any questions. Let's keep

Here’s what students ask on this topic:

How do you calculate the change in velocity from an acceleration-time graph?

To calculate the change in velocity (Δv) from an acceleration-time graph, you need to find the area under the curve. This area represents the change in velocity. For a given time interval, split the area into simple geometric shapes like rectangles and triangles. Use the formulas for the area of a rectangle (base × height) and the area of a triangle (½ × base × height). Sum these areas to get the total change in velocity. If the initial velocity is known, add it to Δv to find the final velocity.

What does the area under an acceleration-time graph represent?

The area under an acceleration-time graph represents the change in velocity (Δv) of an object over a given time interval. This is analogous to how the area under a velocity-time graph represents displacement. By calculating the area under the curve, you can determine how much the velocity has changed during that period.

How do you find the velocity at a specific time using an acceleration-time graph?

To find the velocity at a specific time using an acceleration-time graph, first calculate the change in velocity (Δv) by finding the area under the curve up to that time. If the object starts from rest, the initial velocity (v₀) is zero, so the velocity at that time is simply Δv. If the initial velocity is not zero, add it to Δv to get the final velocity: v = v₀ + Δv.

What is the formula for calculating the area under a curve in an acceleration-time graph?

The formula for calculating the area under a curve in an acceleration-time graph depends on the shape of the area. For a rectangle, the area is base × height. For a triangle, the area is ½ × base × height. By breaking the area under the curve into these simple shapes, you can sum their areas to find the total change in velocity.

How do you handle negative areas in an acceleration-time graph?

In an acceleration-time graph, areas above the time axis represent positive changes in velocity, while areas below the time axis represent negative changes. When calculating the total change in velocity (Δv), subtract the areas below the time axis from the areas above it. This accounts for the direction of the acceleration and ensures the correct calculation of the change in velocity.