graph area

Question

Suppose that you are driving and you gradually accelerate from 50 mph to 70 mph over a two-hour period. Your speed (in miles per hour) hours after you start accelerating can be modeled by the linear function . Find the area under the graph of on the interval and interpret this quantity physically. Explain how you found your answer, and give the correct units for the area.

Answer 1

Mark's comment on the other answer is correct.  The other answer incorrectly identifies the first point on the graph as (1,50).  But it is not going 50 mph after 1 hour; it starts at 50 mph.  So, at time t=0, the speed is 50 mph, and at t = 2 hrs, the speed is 70 mph.  The points are connected by a straight line of the form:

 

70 = 50 + at

 

where

 

a = acceleration (in miles per hour squared, though that does not matter much in this context)

t = hours

 

If you graph velocity on the y axis, and time on the x axis, this plot has y-intercept (0,50), representing the speed and the starting time, and our period of consideration ends at t = 2 hours, where the speed is 70 miles per hour, making the other known point (2,70).  With that information, you could solve for the acceleration (the slope of the graph), if that is what the problem were asking for.

 

Rather, they ask for area under the curve.  As Mark pointed out, this will be a trapezoid (though with its parallel sides vertical, instead of horizontal, as we often see them drawn), and you could find its area with the trapezoid formula.  Or, looking at it in a more conventional way, you could think of it as a (right) triangle with base 2 and height 20 (=70-50) sitting on top of a rectangle of length 2 and height 50.  Either way, you should be able to compute the area.

 

If you would like to check an answer for the area, just let me know.  If you are having trouble with the area calculation, I can definitely help online.

 

As to what that area means, let's imagine a more straightforward case.  Imagine drawing a line on this velocity versus time plot that represents going a constant 60 mph for two hours.  If you go a constant 60 mph for two hours, how far do you travel?  A little dimensional analysis gives:

 

60 (miles/h) * 2 h = 120 miles

 

On a velocity versus time plot, going a constant 60 mph for two hours is drawn as a straight horizontal line at the 60-mph level on the y-axis.  The area under this curve over two hours would be the area of a rectangle with height 60 and width 2.  The area is thus 60 * 2 = 120... exactly the same as the distance covered (or, technically, in physics terms, the displacement).

 

This is a general conclusion, that the area under the curve on a velocity versus time plot represents distance (displacement).  You can also look at it as, when you are finding the area under the curve, just as we did above, you end up multiplying, in some way or another, a velocity and a time.  The product of velocity and time is distance.  So the same conclusion can be reached in that way, as well. 

Answer 2

In the first hour, your speed was 50 mph.  In the 2nd hour, your speed was 70 mph.  We can convert these into coordinate points.

 

Point = (time (in hours), speed (in mph))

 

Point 1 = (1, 50)

Point 2 = (2, 70)

 

Then you will find the slope between the two points.  Then once you find the slope, use the point-slope form of a line to get your linear function.

 

y - y₁ = m(x - x₁)

 

where:

m = slope

(x₁ , y₁) is any of the given points

 

When you find your equation, you can draw a graph to get a visual.  Then you will draw a vertical line from the first endpoint on the graph to the x-axis, and another vertical line from the other endpoint to the x-axis.  You will notice a trapezoid shape formed by this graph.

 

Find the area of the trapezoid to get the area under the curve.  This area under the curve is your acceleration, since your= are changing speed over time.

 

 

Another way to find the area under the curve (if you have taken calculus):

 

Take the definite integral of the linear function from x=1 to x=2.