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 [itex]y= ax^2+ bx+ c[/itex] and you need three equations to solve for the three coefficients, a, b, c. You are given two points, [itex](x_1,y_1)[/itex] and [itex](x_2,y_2)[/itex] on the parabola so [itex]y1= ax_1^2+ bx_1+ c[/itex] and [itex]y2= ax_2^2+ bx_2+ c[/itex]. Those are two of the equations. You also know the "length" of the parabola between [itex]x_1[/itex] and [itex]x_2[/itex]- 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 [itex]y= ax^2+ bx+ c[/itex] and set it equal to that length. That gives you a third equation for a, b, and c.

 

[Architeuthis Dux]

Yes, we can find the equation to a parabola in this scenario. We can use the standard form of a parabola equation, y = ax^2 + bx + c, where a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant term.

To find the specific values of a, b, and c, we can use the two given points and the given time of travel. Since the point travels with uniform velocity, we know that the slope of the parabola at any point is constant. Therefore, we can use the slope formula to find the slope of the line connecting the two given points. This slope will also be the slope of the tangent line to the parabola at the point of intersection.

Next, we can use the point-slope form of a line to find the equation of the tangent line at one of the given points. This equation will have the form y = mx + b, where m is the slope we just calculated and b is the y-intercept.

Since the tangent line and the parabola intersect at the given point, we can set the equations of the tangent line and the parabola equal to each other, and solve for x and y. This will give us two equations with two unknowns (a and b) that we can solve simultaneously to find their values.

Once we have the values of a and b, we can use one of the given points to find the value of c. Plugging these values into the standard form of the parabola equation, we will have the equation to the parabola that satisfies all the given conditions.