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Question

The velocity of a particle is at any time related to the distance travelled by the particle by the relation v=ax+b, where a is positive and b a2. Which of the following statements will be true for this motion? (given x = 0 when t = 0)

A
The displacement of the particle at time t is x=ba(eat1)
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B
The particle will experience a retardation if b < 0
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C
The particle will be at rest at time t = 0
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D
The motion of the particle is under constant acceleration.
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Solution

The correct options are
A The displacement of the particle at time t is x=ba(eat1)
B The particle will experience a retardation if b < 0
v=ax+b;where a+ve and ba2
dxdt=ax+bdxax+b=dt
ln(ax+b)a=t+c (1)
given at t = 0, x = 0,
ln(b)a=c put in(1)
ln(ax+b)a=t+ln(b)a
1aln(ax+bb)=t
ax+bb=eat
x=ba(eat1)
and, v=ax+b=a(ba(eat1))+b=beat
acceleration=abeat
For b<0, acceleration will be negative i.e., particle will experience retardation
v=ax+b=beat
at t = 0, v = b i.e., not at rest.
acceleration =abeat
acceleration is varying with time

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