When a force of constant magnitude and a fixed direction acts on a moving object,then its path is

The coordinates of a particle moving in a plane are given by $x = a \cos(pt)$ and $y = b \sin(pt)$,where $a, b (b < a)$ and $p$ are positive constants of appropriate dimensions. Then:

$A$ particle moves in the $xy$-plane with velocity $v_x = 8t - 2$ and $v_y = 2$. If it passes through the point $(14, 4)$ at $t = 2 \, s$,the equation of its path is:

Two projectiles are fired from the ground with the same initial speeds from the same point at angles $(45^{\circ}+\alpha)$ and $(45^{\circ}-\alpha)$ with the horizontal direction. The ratio of their times of flight is

The displacement vector of a particle of mass $m$ is given by $\vec{r}(t) = A \cos \omega t \hat{i} + B \sin \omega t \hat{j}$.
$(a)$ Show that the trajectory is an ellipse.
$(b)$ Show that $\vec{F} = -m \omega^2 \vec{r}$.

Twelve persons are initially at $12$ corners of a regular polygon of $12$ sides of side length $a$. Each person moves with a uniform speed $v$ in such a manner that person $1$ is always directed towards person $2$,person $2$ towards person $3$,person $3$ towards person $4$,and so on. The time after which they meet is