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Question

For the one-dimensional motion, described by x = t - sint

A
x(t) > 0 for all t > 0
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B
v(t) > 0 for all t > 0
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C
a(t) > 0 for all t > 0
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D
all of these
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Solution

The correct option is A x(t) > 0 for all t > 0
The position of the particle is given as a function of time.
x=tsint

The velocity can be obtained by differentiating the given expression.
v=dxdt=ddt[tsint]=1cost

The acceleration can be obtained by differentiating the expression of velocity w.r.t. time
a=dvdt

a=ddt[1cost]=sint

As acceleration a>0 for all t>0
Hence, x(t)>0 for all t>0

velocity v=1cost
if, cost=1,
the velocity will be v=0

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

Hence, v lies between 0 and 2.

For acceleration
a=dvdt=sint
When t=0;x=0,v=0,a=0
When t=π2;x= positive , v=0,a=1 (negative)
When t=π,x= positive, v= positive , a=0

When t=2π,x=0,v=0,a=0

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