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Math Help - Velocity, Integrals, and Riemann Sums

  1. #1
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    Velocity, Integrals, and Riemann Sums

    Table given:


    Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t=0 seconds. The velocity of the rocket is recorded for selected values of t over the integral 0 \leq t \leq 80 seconds, as shown in the table above.

    a. Find the average acceleration of rocket A over the time interval 0 \leq t \leq 80 seconds.

    b. Explain the meaning of \int_{10}^{70} {v(t)dt} in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate \int_{10}^{70} {v(t)dt}.

    c. Rocket B is launched upward with an acceleration of a(t) = \frac{3}{\sqrt{t+1}} feet per second per second. At time t=0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time t=80 seconds? Explain.

    Thanks a lot to anyone who can help on this!
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  2. #2
    Eater of Worlds
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    One thing you could do to start out is use the given points to find an equation describing the rocket's velocity over time. It is a quadratic.

    I used Excel and got v(t)=\frac{-1}{200}t^{2}+\frac{19}{20}t+5

    Acceleration is the derivative of the velocity. Therefore, the integral of velocity is position.

    Average acceleration is

    \frac{\text{change in velocity}}{\text{time elapsed}}=\frac{v(t_{1})-v(t_{0})}{t_{1}-t_{0}}
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