Question

A particle moves in a straight line such that its displacement, x meters, from a fixed point O on the line at time t seconds is given by 𝑥 = 40[𝑒 −2𝑡 − 𝑒 −4𝑡 ].

(a) Find the time when the particle is instantaneously at rest.

(b) Find the displacement of the particle from O when t = 3 s.

(c) Find the total distance travelled during the first 3 seconds of its motion

Answer 1

Given:

$𝑥 (t)= 40[𝑒^{−2𝑡} − 𝑒^{−4𝑡}]$

(a) Find the time when the particle is instantaneously at rest:

$$x'(t)=40[-2𝑒^{−2𝑡}+4 𝑒^{−4𝑡}]=0$$

$$𝑒^{−2𝑡}(-2+4 𝑒^{−2𝑡})=0$$

$$𝑒^{−2𝑡}=1/2$$

$$t=\frac{1}{2}\ln 2\:\rm s$$

(b) Find the displacement of the particle from O when t = 3 s:

$$d=x(3)=40[𝑒^{−6} − 𝑒^{−12}]=0.1\:\rm m$$

(c) Find the total distance travelled during the first 3 seconds of its motion

$$l=x(3)-x(0)=0.1-0=0.1\:\rm m$$