Ask a class which launch angle sends a ball the furthest and most hands go up for 45 degrees. They are right, but very few can say why, and almost nobody expects the two surprises that come with it. This lesson turns that guess into something students can test, measure and explain in one period.
Start with a prediction, not a formula
Before showing any equation, give the class one question: a ball is launched at 20 m/s. Which angle lands furthest away: 30, 45 or 60 degrees? Ask everyone to vote and write the reason down. A prediction on paper is what makes the result interesting, because now something can be confirmed or overturned.
Then ask a second question that most students get wrong: will 30 degrees or 60 degrees land further?
Run the experiment
Open the projectile motion lab. Set the initial speed to 20 m/s, leave the launch height at 0 m and gravity on Earth. Launch at 45 degrees, then at 30 and 60 without clearing the trails, so all three paths stay on screen together.
Now fill in a table together. These are the values the lab gives at 20 m/s on Earth:
| Launch angle | Horizontal range | Maximum height | Time of flight |
|---|---|---|---|
| 15° | 20.4 m | 1.4 m | 1.06 s |
| 30° | 35.3 m | 5.1 m | 2.04 s |
| 45° | 40.8 m | 10.2 m | 2.89 s |
| 60° | 35.3 m | 15.3 m | 3.53 s |
| 75° | 20.4 m | 19.0 m | 3.94 s |
Surprise one: 30 and 60 degrees land in the same place
Look at the table again. 30 and 60 degrees both land 35.3 m away, and so do 15 and 75 degrees. Students rarely predict this. The two paths look nothing alike: the 60 degree ball climbs three times as high and stays in the air much longer, yet it ends up at the same spot.
The explanation is a trade-off. A steep launch gives a lot of vertical speed, so the ball flies for a long time, but very little horizontal speed to cover ground. A shallow launch is the other way round. Range depends on the product of the two, and angles that add up to 90 degrees give exactly the same product. 45 degrees is where the two are balanced, which is why it wins.
For older students: the range on flat ground is R = v² sin 2θ / g. sin 2θ is largest when 2θ = 90 degrees, and sin 2θ gives the same value for θ and 90 degrees minus θ.
Surprise two: from a cliff, 45 degrees is no longer best
Raise the launch height and the best angle drops below 45 degrees. The ball already has extra time to fall, so it pays to spend more of the speed going forwards. At 20 m/s, the best angle from 10 m up is about 39 degrees, and from 30 m up it is about 32 degrees. Let students find this by trying angles in steps of 5 degrees and recording the range.
This is also a good moment to be honest about the model. The lab ignores air resistance, as most textbook problems do. A real football, javelin or shot put is launched lower than 45 degrees, partly because of the launch height and partly because air drag slows the ball down.
Change the planet
Switch the gravity preset to the Moon. The best angle is still 45 degrees, because gravity affects the vertical and horizontal trade-off in the same way at every angle, but the range grows to about 247 m: roughly six times further, because Moon gravity is about one sixth of Earth gravity.
A lesson plan you can run in one period
- Hook (5 minutes): ask which angle throws furthest and collect votes on the board.
- Prediction (5 minutes): students write down their choice and a reason, then predict whether 30 or 60 degrees goes further.
- Experiment (15 minutes): run 15, 30, 45, 60 and 75 degrees at 20 m/s and fill in the table.
- Explain (10 minutes): turn on the velocity vectors and discuss how the horizontal and vertical parts of the speed change during the flight.
- Extension (10 minutes): raise the launch height or change the planet and find the new best angle.
Misconceptions to watch for
- "The ball stops at the top."Only the vertical speed is zero at the highest point. With the velocity vectors on, students can see the horizontal arrow stay the same length for the whole flight.
- "A heavier ball goes further."Without air resistance, mass does not appear anywhere in the motion. That is why the lab has no mass slider.
- "Higher means further."The 75 degree launch reaches the greatest height in the table but lands as close as the 15 degree one.
Questions to finish with
- Which two angles give a range of 20.4 m at 20 m/s?
- If you double the launch speed, what happens to the range? Predict first, then test it.
- Why do long jumpers take off at a much lower angle than 45 degrees?