Expert: Paul Klarreich - 1/24/2008

Question
at time t=0 , a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly proportional to time t. This brings the jogger to a stop in 10 minutes.

a, write an expression for the velocity of the jogger at time t.

b, what is the total distance traveled by the jogger in that 10-minute interval?

on question a, i got v(t)=300 but i feel there's something missing. will you check if it's right?

Thank you so much.  

Answer
Questioner:   Ann
Category:  Calculus
Private:  No
 
Subject:  calculus
Question:  at time t=0 , a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly proportional to time t. This brings the jogger to a stop in 10 minutes.

a, write an expression for the velocity of the jogger at time t.

b, what is the total distance traveled by the jogger in that 10-minute interval?

on question a, i got v(t)=300 but i feel there's something missing.

>> Yes, I get that same feeling.


will you check if it's right?

Thank you so much.

...............................................

You wrote:
"negative acceleration that is directly proportional to time t"

Any time you have THIS is directly proportional to THAT, write:

THIS = k * THAT.  So write:

x'' = k t

Integrate:

v = x' = kt^2/2 + C1.

OF course, you will need data to get k and C1.

Such as:  V(0) = 300, and  V(10) = 0.   Substitute:


v(0) = C1 = 300, so C1 = 300.

V(10) = k(10^2)/2 + 300 = 0

50k = - 300, so  k = - 6.

Looks like  v(t) = - 3t^2 + 300.

I guess you did leave something out.

For b), you want  x(t).  Integrate again:

x = -3t^3/3 + 300t + C2.
x = - t^3 + 300t + C2.

You won't actually need C2, because this asks for:

x(10) - x(0), so the C2's will drop out.

x(10) = -1000 + 3000 = 2000
x(0)  = 0

Distance = 2000 - 0 = 2000.