Question 90398: I know that my car goes from 0 to 60 mph in 4.2 seconds. How do I determine how many feet the car has traveled in the span. If another car goes to 60 in 5 seconds; how many feet will I beat him by.
Thanks for your help
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The first thing to do is to note that your answer involve feet (how many feet you go and
how many feet you beat him by) ... and seconds.
.
Then note that the speeds reached (60 mph) are inconsistent with your needs. So you need
to convert the mph to feet per second (fps). Do this using dimensional checks. Note that
there are 5280 feet per mile and 3600 seconds per hour. To convert the 60 mph use the
following string:
.
.
Checking the dimensions, note that "miles" in the numerator cancels with "mile" in the denominator
and "hour" in the numerator cancels with "hour" in the denominator. These cancellations
result in the only dimensions left being feet in the numerator and seconds in the denominator
which translates to feet per second or fps.
.
The numbers associated with that come from multiplying 60 times 5280 and dividing that
by 3600. The result of those algebraic manipulations is 88 fps.
.
Now if you graph the point (0, 0) which indicates that at t = 0 your speed is zero, and
(4.2, 88) which indicates that 4.2 seconds your speed is 88 fps, the graph looks something
like this:
.
.
Now trust me on this. The graph forms a right triangle whose three vertices are (0,0),
(4.2,0), and (4.2,88). Since this is a right triangle, one of the legs can be considered
the height of the triangle and the other leg the base. The area of this triangle,
which is the area under the linear graph, is the distance that you travel.
.
The area of a triangle is given by the equation:
.
.
Let's assume that the base is the side along the x-axis, and for this problem, that
length is 4.2 seconds. The height then will be the 88 fps up from the x-axis. Therefore,
the area of the triangle is:
.
.
So in the 4.2 seconds that you accelerate you travel a distance of 184.8 feet. .
If you are into calculus, you may recognize that the equation for this graph is:
.
.
and if you integrate this function with respect to t you get:
.
.
and you evaluate this expression over the range of values from t= 0 to t = 4.2. If you
do you will again get the value of 184.8 feet.
.
Now you can use this same general process to find out how far you competitor (who reaches
88 fps in 5 seconds) goes during the same 4.2 second interval.
.
What we need to do this time is to translate the triangle with vertices (0,0), (0,5), and
(5,88) such that it has the leg along the x-axis reduced to 4.2 seconds. To do that we
will use similar triangles to reduce the size from vertices (0,0), (0,5), and (5,88)
to one that has vertices at (0,0) and (0,4.2). We do this by scaling the triangle using
the ratio 4.2 to 5. Applying this ratio to the vertical side that is 88 units long results
in 
which means that in 4.2 seconds your competitor has reached a speed of 73.92 fps.
This results in the triangle for his performance having the vertices (0,0), (0,4.2),
and (4.2,73.92)
.
.The area of this triangle is:
.
.
So in 4.2 seconds your competitor goes 155.232 feet.
.
[Using calculus you could have gotten this same result by integrating:
.
.
And this results in the same answer of 155.232 feet.
.
So the difference is that in 4.2 seconds you go 184.8 feet and in the same time your
competitor goes 155.232 feet, a difference of 29.568 feet.
.
Hope you can see your way through this problem and what is going on here.
|
|
|
