What determines the nature of the path followed by a particle?

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0 = 10 \, m$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $v = 20 \, m/s$,on a horizontal surface? $(g = 10 \, m/s^2)$

$A$ ball is released from rest from point $P$ of a smooth semi-spherical vessel as shown in the figure. The ratio of the centripetal force and normal reaction on the ball at point $Q$ is $A$,while the angular position of point $Q$ is $\alpha$ with respect to point $P$. Which of the following graphs represents the correct relation between $A$ and $\alpha$ when the ball goes from $Q$ to $R$?

$A$ particle of mass $m$ moves along a circular path of radius $r$ with a time-dependent centripetal acceleration $a_c = k^2rt^2$,where $k$ is a constant. What is the power delivered to the particle by the force acting on it?

$A$ particle is performing uniform circular motion. If $\theta$,$\omega$,$\alpha$,and $a$ are its angular displacement,angular velocity,angular acceleration,and centripetal acceleration respectively,then which of the following is '$WRONG$'? ($v$ is its linear velocity)

$A$ particle of mass $m$ is projected at $45^o$ at $V_0$ speed from point $P$ at $t = 0$. The angular momentum of the particle about $P$ at $t = \frac{V_0}{g}$ is: