Answer to Question #111244 in Mechanics | Relativity for dave

Question

. An object moves along the x axis according to the equation x(t) (3.00t 2 2.00t  3.00) m, where t is in seconds. Determine (a) the average speed between t 2.00 s
and t 3.00 s, (b) the instantaneous speed at t 2.00 s
and at t 3.00 s, (c) the average acceleration between
t 2.00 s and t 3.00 s, and (d) the instantaneous acceleration at t 2.00 s and t 3.00 s.

Answer 1

We have the equation of motion

$x(t)=3t^2+2t+3$

We write expressions for speed

$v(t)=\frac{dx(t)}{dt}= \frac{d(3t^2+2t+3 )}{dt}=6t+2$

We determine the instantaneous velocity at 2s and 3s.

$v(2)=6t+2=6 \cdot 2+2=14 { m/s }$

$v(3)=6t+2=6 \cdot 3+2=20 { m/s }$

The speed has a linear relationship, then the average speed is

$v_{average}=\frac{v(2)+v(3)}{2}=\frac{14+20}{2}=17 m/s$

We write expressions for acceleration

$a(t)=\frac{dv(t)}{dt}= \frac{d(6t+2 )}{dt}=6m/s^2$

To accelerate, we have

$a(2)=a(3)=a_{average}=6 m/s^2$