Projectile motion
Before you start
A basketball shot, the water from a hose and a ball rolling off a table all follow the same kind of curved path. On the Newton’s laws page, a force changes velocity in the direction it acts. Gravity acts only downwards, so it should change only the vertical part of the motion. If that is true, a ball thrown sideways and a ball simply dropped from the same height should land at the same time. That is a prediction we can test here, and the function and quadratic pages give us the tools to describe the resulting curve.
Here we neglect air resistance and treat gravity as a constant downward force. Compare a horizontally thrown ball with one released at the same time from the same height; both begin with zero vertical velocity. The ball travelling farther may seem as though it should stay up longer, but what extra help has it received vertically? Compare their heights at the same instant. Horizontal motion tells us how far the ball travels; vertical motion tells us when it reaches the ground. Keeping both records attached to the same ball and time lets us read the curved path a little at a time.
The setupThe blue projectile and gold comparison ball start at the same height. With launch angle 0°, both initially have zero vertical velocity. Uniform gravity acts downward; air resistance is absent.
What to do and noticeStart with the horizontal launch and press Play. Pause at a few moments and compare the heights of the two balls: they stay level, because both start with no vertical velocity and gravity changes their vertical velocity in exactly the same way, so for both. Then increase the launch speed. The blue ball lands farther away, but at the same moment as before, after seconds. Finally, try a launch angle above 0°: the ball now starts with some upward velocity, so it is no longer level with the dropped ball.
Predict firstWhich lands first: a horizontal launch or a simultaneous drop from the same height?
Use a controlled comparison to understand independent components.
Prerequisites: Velocity has direction and can be resolved into perpendicular components. Near the ground, the gravitational acceleration is about , and it changes only the vertical velocity.Compare with a drop
Predict firstWhich lands first: a horizontal launch or a simultaneous drop from the same height?
Take the ground directly below the launcher as the origin, with positive right and positive upward. Let initial height be and upward speed be . Gravity reduces upward velocity by each second, so after seconds the vertical velocity is .
Velocity changes uniformly, so its average is half the sum of its starting and ending values: . Average velocity times time is displacement, giving . On a velocity–time graph, this is rectangle minus a triangle of base and height .
Both the horizontally launched ball and the dropped comparison have and share and , so their coordinates match at every instant. At landing : , then . Taking the positive time gives . For , , independent of horizontal speed. Launch them and watch both reach the ground together.
How to observe
- Pause at the apex and distinguish vertical speed from total speed
- Set a horizontal launch and compare with a drop from the same height
- Vary speed and gravity to test range and flight time
What this model simplifies
Uniform gravity without air resistance; launch angles are 0° to 85°. Horizontal position is measured from beneath the launch point. Energy is per unit mass in J/kg, with zero potential at ground level.
