Consider a force $\overrightarrow{F}=-x \hat{i}+y \hat{j}$. The work done by this force in moving a particle from point $A(1,0)$ to $B(0,1)$ along the line segment is (all quantities are in $SI$ units).

$A$ time-dependent force $F = 6t$ acts on a particle of mass $1 \ kg$. If the particle starts from rest,the work done by the force during the first $1 \ s$ will be ............... $J$.

$A$ force $F = \alpha + \beta x^2$ acts on an object in the $x$-direction. The work done by the force is $5 \ J$ when the object is displaced by $1 \ m$. If the constant $\alpha = 1 \ N$,then $\beta$ will be: (in $N/m^2$)

$A$ force acts on a $3.0 \ g$ particle in such a way that the position of the particle as a function of time is given by:
$x = 3t - 4t^2 + t^3$
Where $x$ is in metres and $t$ is in seconds. The work done during the first $4 \ s$ is ................. $mJ$

The acceleration-displacement graph of a $10 \ kg$ particle is given. Find the work done as the particle moves from $x = 0 \ cm$ to $x = 8 \ cm$.

$A$ bullet enters a piece of wood with velocity $v_0$ and the resistive force acting on the bullet in the wood is proportional to $v^{\frac{1}{3}}$. If the total distance travelled by the bullet is proportional to $(v_0)^\beta$, then the value of $\beta$ is