SOLUTION: The velocity of a particle moving along the x-axis is v(t) = t^2 – 2t, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance

Algebra ->  Real-numbers -> SOLUTION: The velocity of a particle moving along the x-axis is v(t) = t^2 – 2t, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance       Log On


   

Question 1088682: The velocity of a particle moving along the x-axis is v(t) = t^2 – 2t, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = 3 minutes.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Integrate the velocity.
However since you're looking for the distance travelled and not the total displacement you must take the sign of the velocity into account so you're really integrating the absolute value of the velocity.
s=int%28v%28t%29%2Cdt%29
d=int%28abs%28v%28t%29%29%2Cdt%29
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Looking at the function, the velocity is negative between t=0 and t=2 so break up the integral accordingly,
d=-int%28v%28t%29%2Cdt%2Ct=0%2C2%29%2Bint%28v%28t%29%2Cdt%2Ct=2%2C3%29
So for [0,2],
d%5B1%5D=-t%5E3%2F3%2Bt%5E2%2BC
d%5B1%5D=-2%5E3%2F3%2B2%5E2%2B0%5E3%2F3-0%5E2
d%5B1%5D=-8%2F3%2B4
d%5B1%5D=4%2F3
and from [2,3]
d%5B2%5D=t%5E3%2F3-t%5E2%2BC
d%5B2%5D=3%5E3%2F3%2B3%5E2-2%5E3%2F3%2B2%5E2
d%5B2%5D=4-8%2F3
d%5B2%5D=4%2F3
Then,
d=d%5B1%5D%2Bd%5B2%5D
d=4%2F3%2B4%2F3
d=8%2F3ft
So the total displacement would have been zero but the particle actually travels 4/3 feet in the negative direction and then 4/3 feet in the positive direction.