# Thread: Velocity, Integrals, and Riemann Sums

1. ## 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 $\displaystyle t=0$ seconds. The velocity of the rocket is recorded for selected values of $\displaystyle t$ over the integral $\displaystyle 0 \leq t \leq 80$ seconds, as shown in the table above.

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

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

c. Rocket B is launched upward with an acceleration of $\displaystyle a(t) = \frac{3}{\sqrt{t+1}}$ feet per second per second. At time $\displaystyle t=0$ 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 $\displaystyle t=80$ seconds? Explain.

Thanks a lot to anyone who can help on this!

2. 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 $\displaystyle v(t)=\frac{-1}{200}t^{2}+\frac{19}{20}t+5$

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

Average acceleration is

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