A number of particles are projected from a given point with equal velocities in different directions in the same vertical plane. At any instant they will lie on

Options

A. parabola
B. circle
C. hyperbola
D. rectangle

Explanation

(b) Let a particle is projected at an angle `theta` with horizontal (shown in figure)

At an instant `t`, the particle reaches at point `P(x,y)`.
`:. X=(u cos theta)t`
`:. Ut cos theta =x`………..(i)
and `y=(uyt)-1/2"gt"^(2)`
`:. y=(u sin theta)t- 1/2 "gt"^(2)`
`:. ut sin theta= y+1/2 "gt"^(2)`..........(ii)
Squaring Eqs (i) and (ii) and then adding, we get
`(ut sin theta)^(2)+(ut cos theta)^(2)=x^(2)+(y+ 1/2 "gt"^(2))^(2)`
`:. u^(2)t^(2)=x^(2)+y^(2)+(g^(2)g^(4))/4+"gt"^(2)y`
`x^(2)+y^(2)+(g^(2)t^(4))/4+"gt"^(2)y-u^(2)t^(2)=0`