Prove that the two dimensional motion of a particle in the xy-plane having initial angle θ with the x-axis will be parabolic?

Answer 1

The browser used by this site does not consistently support Greek characters so I will change the angle to w degrees. The direction of the y-axis is taken to be the positive direction so that the upward acceleration due to gravity is -g m/s^2.

Suppose the particle is projected with an initial velocity of u m/s at an angle of t degrees.

Then the vertical component of v is v(y) = v*sin(w) and the horizontal component is v*cos(w).

Suppose the position of the particle after t seconds is [x(t), y(t)].

Then, since there is no force acting in the horizontal direction, x(t) = v*cos(w)*t

or, more simply, x = v*t*cos(w)

this gives t = x/(v*cos(w)

In the vertical direction the only force is gravity so y(t) = v*sin(t) - 1/2*g*t^2

or, more simply, y = v*t*sin(w) - 1/2*g*t^2

substituting for t,

y = v*[x/(v*cos(w)]*sin(w) - 1/2*g*(x/(v*cos(w))^2

or y = x*tan(w) - 1*g/[(2*v^2*cos(w)^2]*x^2

the above equation would look much tidier if only we could write powers as superscript but this crap browser will not allow that.

The equation is, therefore, of the form y = ax^2 + bx for some constants a and b. That is, it is a parabola.