UNIT: CATAPULT

ACT-Based Math: Using the Pythagorean Theorem in Catapult Motion

OBJECTIVE:

Students apply the Pythagorean Theorem to calculate projectile height.

Materials Needed:

Graph paper

Ruler

Calculator

Launch data

Student Directions:

Goal:
You will use the Pythagorean Theorem to calculate the maximum height a projectile reaches during its flight, based on launch data collected from a catapult.

Step 1: Prepare for Launch

Set up your catapult and choose a consistent launch angle (e.g., 45°).

Mark the launch point and set a target area for measurement.

Be ready to record data such as total horizontal distance and time to peak.

Step 2: Launch the Projectile and Measure

Launch the projectile and:

Measure the total horizontal distance (from launch to landing) using a meter stick or measuring tape.

Record the total time of flight using a stopwatch or timer.

Estimate the time to reach peak height by dividing the total flight time by 2.

Example: If the total time of flight is 2 seconds, the time to peak height is 1 second.

Step 3: Calculate the Vertical and Diagonal Components

Use the initial velocity (provided by your teacher or measured separately) and trigonometric functions to find the vertical and diagonal distances traveled at the peak.

Vertical velocity component:
vy=v⋅sin⁡(θ)v_y = v \cdot \sin(\theta)vy​=v⋅sin(θ)

Use this to calculate vertical height using:
h=vy⋅t−12gt2h = v_y \cdot t - \frac{1}{2} g t^2h=vy​⋅t−21​gt2
where g=9.8 m/s2g = 9.8 \, \text{m/s}^2g=9.8m/s2 and ttt is the time to peak.

You can also sketch a right triangle representing the projectile’s motion:

Base = horizontal distance to the peak

Height = maximum vertical height (this is what you're solving for)

Hypotenuse = diagonal distance from launch point to the peak

Step 4: Use the Pythagorean Theorem

If you know the hypotenuse (diagonal distance to the peak) and the horizontal distance, use:

Record your values and compare the calculated height to the one from the vertical motion formula in Step 3.

Step 5: Graph and Visualize

Use graph paper to draw a scale diagram of your projectile’s path.

Label the triangle sides: base (horizontal), height (vertical), and hypotenuse (flight path to peak).

This will help visualize how the components relate to each other.

Step 6: Group Discussion

In your group, discuss the following prompt:

"How do different launch angles affect the height and distance of a projectile?"

Consider how changes in angle affect the balance between vertical and horizontal motion.

Reflect on which angle produced the highest peak and why.