Find the equation to a parabola problem

[srujana_09]

Homework Statement

Can we find the equation to a parabola when two points on it are given,both lying on the same side of the axis of symmetry and also the time of travel between them is given.It is also given that the point travels with uniform velocity along the whole length of the parabola.

 

[HallsofIvy]

Are you assuming that the axis is vertical? If so then any such parabola can be written as $y= ax^2+ bx+ c$ and you need three equations to solve for the three coefficients, a, b, c. You are given two points, $(x_1,y_1)$ and $(x_2,y_2)$ on the parabola so $y1= ax_1^2+ bx_1+ c$ and $y2= ax_2^2+ bx_2+ c$. Those are two of the equations. You also know the "length" of the parabola between $x_1$ and $x_2$- it's the "time of travel" divided by the uniform velocity (I assume you know that velocity- otherwise you do not have enough information to determine the parabola). Write out the equation for the arc-length of $y= ax^2+ bx+ c$ and set it equal to that length. That gives you a third equation for a, b, and c.