Projectile Motion Lab: Analysis and Calculations

Projectile Motion Lab

Introduction

This lab investigates the motion of a projectile launched at various angles, using a projectile launcher, carbon paper, and a stopwatch. The goal is to analyze projectile motion, calculate initial velocities, and predict the range for a given launch angle.

PART I: Determining Initial Velocity Using Horizontal Launch

Equations of Motion

• Horizontal displacement equation:

$y_f = y_i + (v_{yi}) \Delta t$

• For horizontal launches ($v_{yi} = 0$):

$\Delta t = \frac{y_f - y_i}{v_{yi}}$

• Horizontal velocity:

$v_{xi} = \frac{x_f}{\Delta t}$

Experimental Steps

1. Set up the projectile launcher at the edge of a table and select an angle between 0° and 80°.

2. Fire the ball several times to ensure it will land on the carbon paper. Mark the spot where the ball lands.

3. Measure the horizontal distance ($x_f$) from the launcher to the landing spot.

4. Record the data in Table 1.

Table 1: Data for Horizontal Launch

θ	cos θ	xf	Δt	vxi

• Repeat for at least three different angles, each differing by at least 20°.

PART II: Determining Initial Velocity Using Angled Launch

Equations of Motion for Angled Launch

• Vertical displacement equation:

$y_f = y_i + (v_{yi}) \Delta t - \frac{1}{2}g(\Delta t)^2$

• Solving for $v_{yi}$:

$v_{yi} = \frac{\frac{1}{2}g(\Delta t)^2 + y_f - y_i}{\Delta t}$

• Using the sine function to relate $v_{yi}$ and $v_i$:

$\sin \theta = \frac{v_{yi}}{v_i} \implies v_i = \frac{v_{yi}}{\sin \theta}$

Experimental Steps

1. Set up the launcher at a chosen angle and measure the total vertical distance ($y_f$) from the launcher to the floor.

2. Record both the angle and your result in Table 2.

Table 2: Data for Angled Launch

θ	sin θ	xf	Δt	vyi	vi

• Repeat for at least three different angles, each differing by at least 20°.

PART III: Predicting Range and Targeting

Range Formula for Projectile Motion

• When $y_f = y_i = 0$, the range $R$ is given by:

$R = \frac{v_i^2 \sin 2\theta}{g}$

• Solving for θ:

$\theta = \frac{1}{2} \sin^{-1} \left( \frac{gR}{v_i^2} \right)$

Experimental Steps

1. Calculate the average value of $v_i$ from your previous measurements.

2. Measure the horizontal distance $R$ to the target.

3. Use the range formula to calculate the angle θ needed to hit the target.

4. Set the launcher to this angle and attempt to hit the target.

Key Terms and Definitions

• Projectile motion: The motion of an object thrown or projected into the air, subject only to acceleration due to gravity.

• Initial velocity ($v_i$): The velocity at which the projectile is launched.

• Horizontal and vertical components: The initial velocity can be split into horizontal ($v_{xi}$) and vertical ($v_{yi}$) components using trigonometric functions.

• Range ($R$): The horizontal distance traveled by the projectile.

Example Calculation

• Suppose $v_i = 5$ m/s, $g = 9.8$ m/s2, and $R = 2$ m. Then:

$\theta = \frac{1}{2} \sin^{-1} \left( \frac{9.8 \times 2}{5^2} \right) = \frac{1}{2} \sin^{-1} (0.784) \approx 25.8^\circ$

Applications

• Projectile motion principles are used in sports, engineering (e.g., ballistics), and physics research.

Summary Table: Key Equations

Equation	Description
$y_f = y_i + (v_{yi}) \Delta t - \frac{1}{2}g(\Delta t)^2$	Vertical displacement for projectile motion
$x_f = (v_{xi}) \Delta t$	Horizontal displacement for projectile motion
$R = \frac{v_i^2 \sin 2\theta}{g}$	Range of a projectile (when $y_f = y_i$)
$v_{xi} = v_i \cos \theta$	Horizontal component of initial velocity
$v_{yi} = v_i \sin \theta$	Vertical component of initial velocity

Additional info: The lab also emphasizes careful measurement, data recording, and the use of trigonometric relationships to analyze projectile motion. Students are expected to derive key equations and apply them to predict and achieve a target hit, reinforcing both conceptual understanding and practical skills.