Reading an Acceleration Vs. Time Graph

In summary, the conversation discusses finding the velocity of an object with constant acceleration by creating an acceleration vs. time graph. The concept of using the area under the curve to find velocity and distance is explained, and the use of integrals to calculate the exact area is mentioned. The process of using smaller rectangles to approximate the area and then shrinking the width to an infinitely small value is also described. It is noted that if the area is already in the shape of a rectangle, the integral can be skipped and the area can be calculated directly.
  • #1
Yuma
3
0
1. I need to find the velocity of an object with a constant acceleration of 0.5 ft/sec by creating an acceleration vs. time graph spanning the interval of 0 to 5 seconds



2. This is kind of a theoretical problem, so if any equations apply, they were not given to us



3. I have plotted acceleration vs. time, which in this instance is a straight line, and velocity vs. time, which is linear with a constant slope. My question is this: if acceleration is constant, and velocity is the area under that "curve" ( this is a little confusing, as the assignment calls this a curve and it doesn't look like a curve to me!), can I multiply the X coordinate by Y, which =0.5, to get my velocity at various times along this interval? This seems to make sense, as the areas under the "curve" are rectangles given the constant acceleration, but it seems too easy to be right!
 
Last edited:
Physics news on Phys.org
  • #2
When physicists/mathematicians use the term "curve" they often mean any plot on a graph / any continuous line (straight, or curved).
Velocity is the area underneath the acceleration "curve." Similarly, distance is the area underneath the velocity "curve."
To find the velocity at a point in time, you find the area of the given rectangle (from time = 0 until whatever time you are loking for). To find the displacement (position) at a point in time, you find the area of the triangle in the velocity graph.

The process of finding the area under a graph leads to what is called an Integral --> its the precise, mathematical way of finding the area under any shape. Using an integral, you can more easily convert from acceleration to velocity; velocity to distance; or in general convert from the change of something -> to what that thing actual is.
 
  • #3
Yes, you are correct in your method. This is a good intro to basic integral calculus. See if you can follow this:

In your case your acceleration is constant, so the area under the curve is the area of a rectangle.

But what if your acceleration was a(t)=t^2 and you want to know the change in velocity on the time interval [0,5]? Then the acceleration is not a uniform line, so we cannot just find the area using a rectangle.

So how are you going to compute the area under that curve? Well you can still approximate the area under that acceleration curve by making a rectangle, however it's not a very good approximation.

What if instead to split the portions of the graph up into 5 smaller rectangles, each with a width of "1 second". Now if we add these smaller rectangles to get the collective area we will find that they approximate the acceleration a little better.

Now say you split up the rectangles up into 10 rectangles and add them instead of 5, now your approximation is even better. What about adding 50 small rectangles? Now it's a lot better. This is the basic principal of integral calculus.

Here's the leap: What if we shrink the width of those rectangles to an really, really tiny width? Say we make the width of the rectangles infinitely small in their width, now you have an approximation that is so good it's not an approximation anymore, but the actual, exact area under that acceleration curve.

Analyzing graphs that do not have uniform areas becomes somewhat trivial with the help of calculus, this is one reason why it is so valuable.

Using some notation it would look like this:

[tex]\int_{0}^{5}t^2 dt[/tex] Your area is length * width so your length is x^2, and your width is an infinitely small dx. Basically it's the process of summing up a infinite number of infinitely small pieces of area.
 
Last edited:
  • #4
Thanks!

Thanks for helping! I just started doing integrals and Reimann sums a couple of weeks ago in calculus, but the problem I'm working on is for an engineering class. I thought that since we are approximating rectangles to find approximations of the area under the curve in calculus, if the area was rectangular in shape already that I could skip the calculus and just take the area of the rectangle itself. Happily for me, that appears to be the case!

Thanks again!
 
  • #5
Yeah if you can find the area an easier way, normally in the shape of a trapezoid or a rectangle, then you're free to do that, it's the same thing as the integral, just easier.

You took the area 0.5*5

The integral is just

[tex]\int_{0}^{5}0.5 dt = 0.5*5[/tex]
 

1. What is an acceleration vs. time graph?

An acceleration vs. time graph is a graphical representation of the change in acceleration over a specific period of time. It is used to visualize the acceleration of an object and how it changes over time.

2. How do you read an acceleration vs. time graph?

To read an acceleration vs. time graph, you must look at the slope of the line. A steeper slope indicates a greater acceleration, while a flatter slope indicates a slower acceleration. The direction of the slope also indicates the direction of the acceleration.

3. What does the area under the curve represent on an acceleration vs. time graph?

The area under the curve on an acceleration vs. time graph represents the change in velocity over time. This area can be used to calculate the displacement of an object.

4. How does an acceleration vs. time graph differ from a velocity vs. time graph?

An acceleration vs. time graph shows the change in acceleration over time, while a velocity vs. time graph shows the change in velocity over time. A velocity vs. time graph can be used to calculate the acceleration by finding the slope of the line.

5. What are some real-world examples of acceleration vs. time graphs?

Acceleration vs. time graphs can be seen in many real-world scenarios, such as a car accelerating from a stop, a roller coaster going up and down hills, or a person jumping off a diving board. These graphs can also be used to analyze the motion of objects in sports, such as a basketball player sprinting down the court or a skier going down a slope.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
25
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
4K
  • Introductory Physics Homework Help
Replies
12
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
201
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
942
  • Introductory Physics Homework Help
Replies
1
Views
7K
Back
Top