A parabolic trajectory is the curve described by an object thrown in a uniform gravitational field, neglecting air friction. It is the combination of uniform horizontal motion and vertical free fall. This concept is part of the senior high school curriculum.
Discover FizziQ
How to measure it in class
With the FizziQ app, video tracking allows precise analysis of a projectile’s trajectory.
Steps:
- Film a ball throw with FizziQ (side view, fixed camera)
- Open the video in the tracking module
- Mark the ball’s position frame by frame
- Export the data (x, y, t)
- Verify that x(t) is linear (uniform horizontal motion)
- Verify that y(t) is parabolic (uniformly accelerated vertical motion)
- Calculate g from the curvature of y(t)
Scientific activities on this topic
- Tennis ball trajectory: https://www.fizziq.org/en/activities/parabolic-flight/
- Calculation of g by video analysis
- Study of range as a function of launch angle
Learn more
Equations of motion:
Choosing the origin at the launch point and the y-axis pointing upward:
- x(t) = v0 x cos(alpha) x t (uniform motion)
- y(t) = v0 x sin(alpha) x t - (1/2)gt^2 (uniformly accelerated motion)
where v0 is the initial velocity and alpha is the launch angle relative to the horizontal.
Trajectory equation:
By eliminating t, we obtain: y = tan(alpha) x x - g x x^2 / (2v0^2cos^2(alpha))
This is indeed the equation of a parabola (y = ax + bx^2).
Maximum range:
The range (horizontal distance reached) is maximum for an angle of 45 degrees (in the absence of friction). It then equals: x_max = v0^2/g
Effect of friction:
In the presence of air friction, the trajectory is no longer exactly parabolic: it is asymmetric, with the descent being steeper than the ascent.
Formula
Equations of motion (without friction):
x(t) = v0 x cos(alpha) x t y(t) = v0 x sin(alpha) x t - (1/2) x g x t^2
Trajectory equation: y = x x tan(alpha) - g x x^2 / (2 x v0^2 x cos^2(alpha))
Range: R = v0^2 x sin(2alpha) / g
Maximum height: H = v0^2 x sin^2(alpha) / (2g)
Application examples
- A basketball player shoots at 10 m/s with an angle of 45 degrees: range approximately 10 m
- Water from a fountain jet describes a parabola
- A cannonball in the 17th century followed a parabolic trajectory (Galileo’s model)
- Long jump: the athlete describes a parabola in the air
FAQ
Q: Why do we say it’s a parabola and not an ellipse? A: It’s an approximation valid for short trajectories where the gravitational field is uniform. On a planetary scale, trajectories are ellipses (Kepler’s laws).
Q: How can I measure the launch angle with FizziQ? A: By marking the first positions of the projectile, calculate the initial slope dy/dx = tan(alpha).
Q: Do golf balls follow a parabola? A: Not exactly, because friction and the Magnus effect (spin) significantly modify the trajectory. A golf ball with spin can rise above the theoretical parabolic trajectory.
Q: How can I verify that I get a parabola? A: By plotting y as a function of x^2, you should get a straight line (after adding a linear term for the x term).
Related concepts
Kinematics - Free fall - Gravitational acceleration - Video tracking - Motion in a uniform field - Ballistics