HW question Please Help!!

[mathhelp418]

Hello. I have problem that says...A car accelerates smoothly from 0 to 60 mph in 10 seconds with the velocity given in the figure below. Estimate how far the car travels during the 10 second period. I am given a graph that has velocity (mph) on the y axis and time (sec) on the x axis with an upward sloping curve. I understand that I need to take the area under the curve and I know that would be 1/2(b*h) --> 1/2(60*10) ...but I do not know how to answer the question which says...The care travels ______ feet. I don't know how to get to the feet part. Thanks!!

 

[JeffM]

Hello. I have problem that says...A car accelerates smoothly from 0 to 60 mph in 10 seconds with the velocity given in the figure below. Estimate how far the car travels during the 10 second period. I am given a graph that has velocity (mph) on the y axis and time (sec) on the x axis with an upward sloping curve. I understand that I need to take the area under the curve and I know that would be 1/2(b*h) --> 1/2(60*10) ...but I do not know how to answer the question which says...The care travels ______ feet. I don't know how to get to the feet part. Thanks!!

Does "smoothly" mean "linearly"? We can't see your graph.

From your work, I am guessing that \(\displaystyle v = 6t.\) Is that correct?

\(\displaystyle Let\ D = distance.\)

\(\displaystyle So\ \dfrac{dD}{dt} = v = 6t \implies D = \int v\ dt = \int 6t\ dt \implies D = 3t^2 + K.\)

\(\displaystyle But\ D = 0\ if\ t = 0 \implies K = 0.\)

\(\displaystyle So\ D = 3t^2 \implies D = 3 * 10^2 = 3 * 100 = 300.\)

So if the function of v is linear, meaning constant acceleration, you got the right answer by using geometry. You just did not realize it. You would have realized it had you solved the problem using calculus. If the graph of v is not linear, you may still be able to solve it by calculus even if plane geometry lets you down.

 

[mathhelp418]

Thank you! The only thing that I still don't understand is that they want the answer in feet, but the y axis is in miles per hour and the x axis is in seconds so If I have 300 mph/sec how would I convert that to feet? Thanks!

 

[JeffM]

Thank you! The only thing that I still don't understand is that they want the answer in feet, but the y axis is in miles per hour and the x axis is in seconds so If I have 300 mph/sec how would I convert that to feet? Thanks!

You don't.

You get your units straight before you do anything else.

Restate the function for velocity into feet per second from miles per hour. Now you have comparable units along both axes.

By the way, I still do not know (because you still have not told me) whether the function for velocity is linear.