(C) Given acceleration $a = \frac{dv}{dt} = bt$.Integrating with respect to time $t$:$\int dv = \int bt \, dt \Rightarrow v = \frac{1}{2}bt^2 + C_1$.A

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(C) Given acceleration $a = \frac{dv}{dt} = bt$.Integrating with respect to time $t$:$\int dv = \int bt \, dt \Rightarrow v = \frac{1}{2}bt^2 + C_1$.At $t = 0$,$v = v_0$,so $C_1 = v_0$.Thus,the velocity is $v = \frac{1}{2}bt^2 + v_0$.Since $v = \frac{dx}{dt}$,we have $\frac{dx}{dt} = \frac{1}{2}bt^2 + v_0$.Integrating with respect to time $t$:$x = \int (\frac{1}{2}bt^2 + v_0) dt = \frac{1}{2}b(\frac{t^3}{3}) + v_0t + C_2$.At $t = 0$,$x = 0$,so $C_2 = 0$.Therefore,the distance travelled is $x = v_0t + \frac{1}{6}bt^3$.

Class 11 Physics Motion in Straight Line Uniformly Accelerated Motion

Medium

The 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
${v_0}t + \frac{1}{3}b{t^2}$
B
${v_0}t + \frac{1}{3}b{t^3}$
C
${v_0}t + \frac{1}{6}b{t^3}$
D
${v_0}t + \frac{1}{2}b{t^2}$

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Motion in Straight Line

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Uniformly Accelerated Motion

$A$ body starts from rest and acquires a velocity of $10 \ m \ s^{-1}$ in $2 \ s$. What is the acceleration of the body and the distance travelled?

MediumTS EAMCET 2022

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Give examples of a one-dimensional motion where:
$(a)$ The particle moving along positive $x$-direction comes to rest periodically and moves forward.
$(b)$ The particle moving along positive $x$-direction comes to rest periodically and moves backward.

Medium

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An engine of a train,moving with uniform acceleration,passes a signal post with velocity $u$ and the last compartment passes the same signal post with velocity $v$. The velocity with which the middle point of the train passes the signal post is

MediumJEE MAIN 2021

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$A$ body of mass $4 \,kg$ is accelerated by a constant force. It travels a distance of $5 \,m$ in the first second and a distance of $2 \,m$ in the third second. The force acting on the body is: (in $\,N$)

EasyKCET 2008

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$A$ particle initially at rest moves along the $x$-axis. Its acceleration varies with time as $a = 4t$. If it starts from the origin,the distance covered by it in $3 \ s$ is $........... \ m$.

Medium

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The 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

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$A$ body starts from rest and acquires a velocity of $10 \ m \ s^{-1}$ in $2 \ s$. What is the acceleration of the body and the distance travelled?

Give examples of a one-dimensional motion where:$(a)$ The particle moving along positive $x$-direction comes to rest periodically and moves forward.$(b)$ The particle moving along positive $x$-direction comes to rest periodically and moves backward.

An engine of a train,moving with uniform acceleration,passes a signal post with velocity $u$ and the last compartment passes the same signal post with velocity $v$. The velocity with which the middle point of the train passes the signal post is

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Give examples of a one-dimensional motion where:
$(a)$ The particle moving along positive $x$-direction comes to rest periodically and moves forward.
$(b)$ The particle moving along positive $x$-direction comes to rest periodically and moves backward.