For the one-dimensional motion, described by x = t - sint, the following statements are given.

(a) x(t) > 0 for all t > 0
(b) v(t) > 0 for all t > 0
(c) a(t) > 0 for all t > 0
(d) v(t) lies between 0 and 2

Choose the correct option:

1. (a, c)

2. (b, c)

3. (a, d)

4. (b, d)

(3) Hint: The first derivative of x gives velocity and the first derivative of velocity gives acceleration.

Step 1: Find the velocity and acceleration.

Given,

x=tsint velocity v=dxdt=ddt[tsint]=1cost Acceleration a=dvdt=ddt[1cost]=sint

Step 2: Put the different values of t to find the correct answer.

As acceleration             a > 0 for all t > 0
Hence,                     x(t) > 0 for all t > 0
                       Velocity  v= 1 - cos t
When                cos t = 1, velocity v = 0

Vmax=1(cost)min=1(1)=2vmin=1(cost)max=11=0

Hence, v lies between 0 and 2.

 Acceleration a=dvdt=sint

 When t=0;x=0,x=+1,a=0 When t=π2;x=1,v=0,a=1 When t=π;x=0,x=1,a=1 When t=2π;x=0,x=0,a=0