Question

1. a particle moves according to the equation $x = 10t^2$, where $x$ is in meters and $t$ is in seconds. (a) find the average velocity for the time interval from 2.00 s to 3.00 s. (b) find the average velocity for the time interval from 2.00 to 2.10 s.

Explanation:

Step1: Define average velocity formula

Average velocity $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x(t_2)-x(t_1)}{t_2-t_1}$

Step2: Solve part (a): Calculate $x(3.00)$

$x(3.00) = 10(3.00)^2 = 10 \times 9 = 90$ m

Step3: Solve part (a): Calculate $x(2.00)$

$x(2.00) = 10(2.00)^2 = 10 \times 4 = 40$ m

Step4: Solve part (a): Compute average velocity

$\bar{v} = \frac{90 - 40}{3.00 - 2.00} = \frac{50}{1} = 50$ m/s

Step5: Solve part (b): Calculate $x(2.10)$

$x(2.10) = 10(2.10)^2 = 10 \times 4.41 = 44.1$ m

Step6: Solve part (b): Compute average velocity

$\bar{v} = \frac{44.1 - 40}{2.10 - 2.00} = \frac{4.1}{0.1} = 41$ m/s

Answer:

(a) 50 m/s
(b) 41 m/s