Question

Q2. The altitude of a rocket in the first half-minute of its ascent is given by x = bt2, where x is the altitude and b = 2.90 m/s2 is a constant.
a) Find a general expression for the rocket’s velocity as a function of time.
b) Find the rocket’s instantaneous velocity at t = 20 s.
c) Find an expression for the average velocity of the rocket. Find the average velocity after during the first 20 seconds.
d) Compare your two velocities (instantaneous and average). Is there a difference and if so why?

Answer 1

Q2. The altitude of a rocket in the first half-minute of its ascent is given by $x = bt2$, where $x$ is the altitude and $b = 2.90 \, \text{m/s}^2$ is a constant.

a) Find a general expression for the rocket's velocity as a function of time.

b) Find the rocket's instantaneous velocity at $t = 20 \, \text{s}$.

c) Find an expression for the average velocity of the rocket. Find the average velocity after during the first 20 seconds.

d) Compare your two velocities (instantaneous and average). Is there a difference and if so why?

Solution:

Such as $v = \frac{dX}{dt}$

$$v = (bt^2)' = 2bt$$ $$v = 2.9 * 2t = 5.8t$$ $$v(t = 20 \, \text{s}) = 5.8 * 20 = 116 \, \text{m/s}$$ $$v(\text{average}) = \frac{v_0 + v_t}{2}$$ $$v(\text{average} \, t = 20) = \frac{v_0 + v(t = 20)}{2} = \frac{0 + 116}{2} = 58 \, \text{m/s}$$

Answers: $v = 5.8t$, $v = 116 \, \text{m/s}$, $v(\text{average}) = \frac{v_0 + v_t}{2}$, $v(\text{average} \, t = 20) = 58 \, \text{m/s}$

The instantaneous and average velocities are different.