Kinematics

Question

A particle P moves in a straight line so that , after t seconds, its displacement, x metres, from a fixed point O is given by the equation x=12sin(2t)-6 . Determine

 

(a) expressions for the velocity and acceleration of P in terms of t,

(b) the time at which P first return to O

(c) the maximum distance of P from O during the motion,

(d) the speed of P when x=3.

Answer 1

a)

x(t) = 12sin(2t)-6

v(t) = x'(t) = 24cos(2t)

a(t) = v'(t) = x''(t) = -48sin(2t)

 

b)

x(t) = 12sin(2t)-6

x(0) = 12sin(0)-6 = -6

12sin(2t)-6=-6

12sin(2t)=0

sin(2t)=0

2t=pi

t=pi/2

 

c)

Sin(2t) has a maximum value of 1 which it reaches at t=pi/4. That means the maximum distance from O that the particle will go then is x(pi/4) - x(0) = 6--6 = 12 meters

 

d)

12sin(2t)-6=3

12sin(2t)=9

sin(2t)=3/4

t=.424 radians

v(t)=24cos(2t)

v(.424)=24cos(.848)= 15.87 m/s