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Question

Acceleration of a particle is increasing linearly with time $$t$$ as $$bt$$. The particle starts from the origin with an initial velocity $${ v }_{ 0 }$$. The distance travelled by the particle in time $$t$$ will be


A
v0t+bt36
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B
v0t+bt23
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C
v0t+bt33
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D
v0t+bt22
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Solution

The correct option is C $${ v }_{ 0 }t+\cfrac { b{ t }^{ 3 } }{ 3 } $$
$$a=bt\\ \dfrac { dv }{ dt } =bt\\ dv=b tdt$$
Integrating both sides, we get
$$v-{ v }_{ o }=\dfrac { b{ t }^{ 2 } }{ 2 } \\ v={ v }_{ o }+\dfrac { b{ t }^{ 2 } }{ 2 } \\ \dfrac { dx }{ dt } ={ v }_{ o }+\dfrac { b{ t }^{ 2 } }{ 2 } \\ dx=({ v }_{ o }+\dfrac { b{ t }^{ 2 } }{ 2 } )dt$$
Integrating both sides, we get
$$\Delta x={ v }_{ o }t+\dfrac { b{ t }^{ 3 } }{ 6 } $$

Physics

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