# How to know initial jumping velocity if I know trajectory as parabola equation and gravity value?

I have a parabola equation $$y = ax^2 + bx + c$$ with vertex at (vertexX, vertexY)) which draws the object's moving trajectory (from point $$A(0;0)$$ to $$B(X;Y)$$). Also, I have a gravity vector ($$G(0;-g)$$).

How to find the initial velocity which I should to set to this object to move it by this trajectory?

I tried the (vertexY / b, vertexY) but it is too small and doesn't depend on gravity.

Let initial velocity be $$(u_x,u_y)$$

differentiating the parabolic equation w.r.t. x gives $$\frac{dy}{dx} = b$$ when $$x=0$$

so $$\frac{u_y}{u_x} = b$$

Also from the equations of motion horizontally and vertically using time $$t$$ to be the time to the vertex $$(X,Y)$$

$$t=\frac{X}{u_x}$$

$$Y = u_yt -\frac{gt^2}{2}$$

solving these three brings us to a suggestion for your initial velocity of

$$u_x = X(\frac{2}{g}(bX-Y))^{-0.5}$$

and $$u_y = bu_x$$

hope that works...

• No, it doesn't work. Velocity is too small, bots even doesn't stop to touch ground Oct 1, 2021 at 14:22
• @ Robotex could you give an example with $(X,Y)$ and $b$ and the value of $g$ used, let's put the numbers in and see if we are getting the same $u_x$, $u_y$ Oct 1, 2021 at 14:28
• Oh, sorry, I made a mistake in formula. Looks like, that is working. Thank you :) Oct 1, 2021 at 14:35
• @ Robotex OK, glad it helped Oct 1, 2021 at 14:36
• @ Qmechanic the tag edit seems a bit late, the question has been live for 3 hours... Oct 1, 2021 at 14:43

First you have $$x(0)=y(0)=0$$. $$v_y/0)/v_x(0)=b$$, second you have $$v_y(vertex)=0, and , v_y(t)=v_y(0)-g*t , y(vertex)=v_y(0)*t-g/2*t^2, x(t)=v_x(0)*t$$ this should be enough to find a,b,c

• I have a, b and c. I need to find V0 Oct 1, 2021 at 14:07

This formula gives approximately correct result (but with longer time than I put to formula):

Velocity = FVector( ((vertY / Props.JumpTime) + (0.5f * -GravityZ * Props.JumpTime)) / b, 0.0f, (vertY / Props.JumpTime) + (0.5f * -GravityZ * Props.JumpTime) );

VertY - the Y of highest point of parabola GravityZ - gravity magnitude Props.JumpTime - time for moving