# Properties of Parabolas ### Fun Properties of Parabolic Trajectories

As we have found, the shape of the trajectory of an object moving in 2 dimensions subject to an acceleration in one direction is a parabola.

Splitting the motion problem into separate one dimensional problems, with the acceleration in the vertical direction gives:

S x

= V ox

t

V x

= V ox

S y

= V oy

t – ½ gt

V y

= V oy

–gt

2

Shortening S x

to x and S y

to y and combining the 2 expressions gives x = V ox

t --> t = x/V ox y = V oy

t – ½ gt 2 = V oy

x/V ox

– ½ g(x/V ox

) 2 y = (V oy

/V ox

) x – ½ (g/V ox

2 ) x 2 , the equation for the parabolic path.

 The acceleration is constant, g downwards for the whole time.

 The vertical component at maximum height is 0.

 The vertical part of the motion is identical to the motion of a projectile tossed vertically upward with V oy

.

If air resistance is negligible, and the initial and final heights are the same, there are some symmetries which can be useful:

 The time to reach maximum height equals half the total time of flight.

 The final speed equals the initial speed.

 V y

(final) = - V oy

 At any given height, the speed of the projectile is the same and

V y

(down) = - V oy

(up)

 The range at any angle θ is equal to the range for angle (90 ° -θ), or the ranges are equal for angles (45 ° -θ) and (45 ° +θ).

A further curious property of these trajectories is that, if the initial velocity is held constant and the trajectories for all possible angles are plotted, the curve which encloses all these trajectories and is tangent to each at a single point is a parabola.