Calculus story problem (unit on minimal paths, distance, and displacement)?
Olive Oyl is on a conveyor belt moving 3 ft/s toward the sawmill. At time t=0, Popeye rescues her and starts running in the other direction along the conveyor belt. His velocity with respect to the ground, v, in ft/s, is given by
v= sqrt(t) - 3
When does Popeye's velocity become positive? How far have he and Olive moved toward the sawmill at this time? What is their net displacement from the starting point at t = 25? What total distance did they go from t=0 to t = 25?
The first part of the question is simple enough. The square root of t has to just cancel the 3 ft/s in the opposite direction. So is has to be 9 seconds. To find out the total distance traveled just integrate the equation over the interval:
Integral from 0 to 9 of sqrt(t) - 3 = -9 feet. This makes sense because we know Popeye has been losing ground the whole time, we should get a negative number.
To finish off the problem integrate the same equation from 9 to 25.
Integral from 9 to 25 of sqrt(t) - 3 = 52 / 3, or about 17.3 feet. Since they traveled back 9 feet first their net displacement is about 8.3 feet. The total distance traveled though, is 17.3 feet plus 9 feet, or about 26.3 feet.
Lets start
1.
His velocity is negative at start. The change will occur at v=0
0 = square root (t) - 3
square root (t) = 3
t = 3^2
t = 9s ————————->ANSWER
2.
At this time his relative velocity wrt earth is zero so he is just moving with the belt
S = ut
S = 3*9
S = 27 ft ———————->ANSWER
3.
displacement is on the ground ; use his velocity on the ground
S = v * t
S = { square root(25) - 3 } * 25
S = 2 * 25
S = 50 ft ———————>ANSWER
4.
distance is on the belt so use the velocity w.r.t. belt
v(p,e)= {sqrt(t) - 3}
v(b,e)= -3ft/s because movement is other direction
v(p,b) = v(p,e) + v(e,b)
v(p,b) = {sqrt(t) - 3} + 3
= sqrt(25) -3 +3
= 5 ft/s
S = ut
S = 5 * 25
S = 125 ft ———————–>ANSWER