Example 83 A particle is moving in a straight line underacceleration a

1.	Example 83 A particle is moving in a straight line underacceleration a = kt, where k is a constant. Find the velocityin terms of t. The motion starts from rest.
Answer» GIVEN, the acceleration, $\longrightarrow\sf{a(t)=kt}$ But we know that acceleration is the FIRST DERIVATIVE of VELOCITY with respect to time. $\longrightarrow\sf{a(t)=\dfrac {d}{dt}[v(t)]}$ Then, $\longrightarrow\sf{\dfrac {d}{dt}[v(t)]=kt}$ $\longrightarrow\sf{d[v(t)]=kt\ dt}$ $\displaystyle\longrightarrow\sf{v(t)=\int kt\ dt}$ $\displaystyle\longrightarrow\sf{v(t)=k\int t\ dt}$ $\displaystyle\longrightarrow\sf{v(t)=k\cdot\dfrac {t^2}{2}+c}$ $\displaystyle\longrightarrow\sf{v(t)=\dfrac {1}{2}kt^2+c}$ Since the MOTION starts from rest, $\displaystyle\sf{v=0}$ at $\displaystyle\sf{t=0.}$ Thus, $\displaystyle\longrightarrow\sf{v(0)=0}$ $\displaystyle\longrightarrow\sf{\dfrac {1}{2}k(0)^2+c=0}$ $\displaystyle\longrightarrow\sf{c=0}$ Therefore, $\displaystyle\longrightarrow\sf{\underline {\underline {v(t)=\dfrac {1}{2}kt^2}}}$