Question

A particle moves along x-axis according to expression x=3t²-2t where x is in metre and t in seconds.

a) What is the displacement of the particles in the time interval

t=0 to t=1

t=1 to t=4

b) Calculate the average velocity for the time intervals in (a)

c) Find the instantenous velocity for a particle at 3 seconds

Answer 1

Given:

$x(t)=3t²-2t$

a) the displacement of the particles

$$d_{01}=x(1)-x(0)=3*1^2-2*1-0=1\:\rm m$$

$$d_{14}=x(4)-x(1)\\=(3*4^2-2*4)-(3*1^2-2*1)=39\:\rm m$$

b) the average velocity of the particle

$$v_{01}=\frac{d_{01}}{\Delta t}=\rm\frac{1\: m}{1\: s}=1\: m/s$$

$$v_{14}=\frac{d_{14}}{\Delta t}=\rm\frac{39\: m}{3\: s}=13\: m/s$$

c) the instantaneous velocity

$$v(t)=x'(t)=(3t^2-2t)'=6t-2$$

at the instant $t=3\:\rm s$

$$v(3)=6*3-2=16\:\rm m/s$$