A particle moving along $x$-axis has acceleration $f$, at time $t$, given by $f=$…
A particle moving along $x$-axis has acceleration $f$, at time $t$, given by $f=$ $f_0\left(1-\frac{t}{T}\right)$, where $f_0$ and $T$ are constants.
The particle at $t=0$ has zero velocity. In the time interval between $t=0$ and the instant when $f=0$, the particle's velocity $\left(v_x\right)$ is:
$\frac{1}{2} f_0 T^2$
$f_0 T^2$
$\frac{1}{2} f_0 T$
$f_0 T$
Solution
Acceleration $\frac{d v}{d t}=f=f_0\left(1-\frac{t}{T}\right)$
$\begin{aligned}
\Rightarrow{ }_0^v d v & =f_0^T 1-\frac{t}{T} d t \\
\Rightarrow \quad v & =f_0\left(t-\frac{t^2}{2 T}\right)_0^T \\
& v =f_0\left(T-\frac{T^2}{2 T}\right)=\frac{1}{2} f_0 T
\end{aligned}$