Optimal launch angle
Discover the launch angle that maximizes a projectile's range and observe the symmetry of trajectories with the FizziQ Web Ballistics simulation.
If you had to throw a ball as far as possible, at what angle would you launch it? Straight ahead? At 60°? At 45°? Artillery officers have sought the answer for centuries. The FizziQ Web Ballistics simulation lets you fire projectiles at every angle from 10° to 80° and compare the resulting ranges. You will discover that 45° gives the maximum range, that complementary angles (like 30° and 60°) give equal ranges, and understand the physics behind these elegant results.
Learning objectives:
The student uses the FizziQ Web Ballistics simulation to launch projectiles at different angles while keeping the same initial speed. They record the range for each angle and plot the range versus angle graph to identify the optimal angle and the symmetry.
Level:
High school
FizziQ Web
Author:
Duration (minutes) :
30
What students will do :
- Identify the launch angle that maximizes projectile range
- Observe the symmetry of trajectories for complementary angles
- Record data in the FizziQ experiment notebook and plot range versus angle
- Verify the formula R = v₀² sin(2α)/g
- Understand the interplay between horizontal and vertical velocity components
Scientific concepts:
- Parabolic motion
- Projectile range
- Optimal launch angle (45°)
- Independence of horizontal and vertical motions
- Complementary angles and symmetry
Sensors:
- FizziQ Web Ballistics simulation
What is required:
- Computer, tablet, or smartphone with FizziQ Web
Experimental procedure:
Open FizziQ Web. In the sidebar, click Experiment, then Simulations, and select Ballistics Simulation.
Set the initial speed to 20 m/s. Disable air resistance to start. These parameters will remain fixed throughout.
Set the launch angle to 10°. Press REC to fire and record the data. The trajectory appears on screen. Note the range in the experiment notebook.
Repeat for angles 20°, 30°, 40°, 45°, 50°, 60°, 70°, and 80°. Each trajectory appears in a different color on the same screen.
Observe the superimposed trajectories. Which angles seem to give the same range? Note your observations.
For each shot, read the range (horizontal distance at impact) from the experiment notebook.
Create a table with two columns: Angle (°) and Range (m). Enter your measurements.
Plot the graph of range versus angle. For which angle is the range maximum?
Compare the ranges for 30° and 60°, for 20° and 70°, for 10° and 80°. What do you notice?
State a conclusion: what is the optimal angle and why do complementary angles give the same range?
Expected results:
The range-angle graph has a bell-shaped curve, symmetric about 45°. Maximum range is achieved at 45°. Complementary angles (e.g., 30° and 60°) give exactly the same range. For v₀ = 20 m/s: R_max = v₀²/g ≈ 40.8 m at 45°. The formula R = v₀² sin(2α)/g explains the symmetry since sin(2×30°) = sin(2×60°) = sin(60°) = sin(120°).
Scientific questions:
- Why does the 45° angle give the maximum range?
- If two complementary angles give the same range, are the trajectories identical? What differs?
- How does the initial speed affect the maximum range?
- What would the optimal angle be if the launch point is higher than the landing point?
- How does air resistance change the optimal angle?
- Why is sin(2α) the key function in the range formula?
Scientific explanations:
Without air resistance, a projectile's trajectory is a parabola. The motion decomposes into two independent motions: uniform horizontal motion (vₓ = v₀ cos α) and uniformly accelerated vertical motion (aᵧ = -g).
The range R of a projectile launched from ground level with speed v₀ at angle α is: R = (v₀² × sin(2α)) / g.
The function sin(2α) has the property that sin(2α) = sin(180° - 2α). This implies that sin(2×30°) = sin(2×60°), which explains why complementary angles give equal ranges.
At 45°, the projectile shares its kinetic energy equally between horizontal and vertical components. This is the optimal compromise between range (favored by high horizontal speed) and hang time (favored by high vertical speed).
A low angle (10°) gives a projectile that is fast horizontally but falls quickly. A high angle (80°) gives a projectile that goes high but covers little horizontal distance. 45° is the perfect balance.
Extension activities:
- Why does the 45° angle give the maximum range?
- If two complementary angles give the same range, are the trajectories identical? What differs?
- How does the initial speed affect the maximum range?
- What would the optimal angle be if the launch point is higher than the landing point?
- How does air resistance change the optimal angle?
- Why is sin(2α) the key function in the range formula?
Frequently asked questions:
Q: The ranges for 30° and 60° are not exactly equal.
R: Check that air resistance is disabled. With air resistance, the symmetry is broken and the optimal angle drops below 45°.
Q: The maximum range does not match v₀²/g.
R: The formula R = v₀²/g applies at 45° only. For other angles, use R = v₀² sin(2α)/g. Also check that the launch height equals the landing height.
Q: Why are all the trajectories parabolas?
R: A parabola results from the combination of uniform horizontal motion and uniformly accelerated vertical motion. This is always the case without air resistance.
Q: What is the optimal angle with air resistance?
R: It depends on the projectile's mass, size, and speed, but typically falls between 30° and 42° for common projectiles.