Projectile Motion Lab: Study Notes and Analysis

Projectile Motion Lab

Introduction to Projectile Motion

Projectile motion is a fundamental concept in physics that describes the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. This lab explores the kinematics of projectile motion by measuring the range and initial velocity of a projectile launched at various angles.

• Projectile: An object upon which the only force acting is gravity after it is launched.

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

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

PART I: Determining Initial Velocity for Horizontal Launch

Equations and Setup

For a projectile fired horizontally, the following kinematic equation is used:

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

Since the initial vertical velocity $v_{y_i} = 0$, the equation simplifies to:

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

Key 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 and use carbon paper to mark where it lands.

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

4. Record the time of flight ($\Delta t$) and calculate the initial velocity using the equation:

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

Table 1: Data Collection for Horizontal Launch

The table below is used to record the angle, cosine of the angle, horizontal distance, time of flight, and calculated initial velocity.

θ	cos θ	xf	Δt	vx_i

PART II: Determining Initial Velocity for Angled Launch

Equations and Derivations

For projectiles fired at an angle, the vertical displacement equation is:

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

Assuming $y_i = 0$, rearrange to solve for $v_{y_i}$:

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

Key Steps:

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

2. Record the time of flight ($\Delta t$).

3. Calculate the initial vertical velocity ($v_{y_i}$) using the derived equation.

4. Use the sine function to relate $v_{y_i}$ to the total initial velocity:

$\sin \theta = \frac{v_{y_i}}{v_i}$

Therefore,

$v_i = \frac{v_{y_i}}{\sin \theta}$

Table 2: Data Collection for Angled Launch

θ	sin θ	xf	Δt	vy_i	vi

PART III: Calculating Range and Targeting

Range Formula and Application

When the initial and final vertical positions are equal ($y_i = y_f = 0$), the range of the projectile is given by:

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

Key Steps:

• Calculate the average initial velocity ($v_i$) from previous measurements.

• Measure the horizontal distance ($R$) to the target.

• Use the range formula to solve for the angle ($\theta$) needed to hit the target:

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

Example Application

• Suppose $v_i = 5$ m/s and $R = 2$ m. Plug these values into the formula to solve for $\theta$.

• Set the launcher to the calculated angle and attempt to hit the target.

Summary Table: Key Equations

Equation	Description
$y_f = y_i + (v_{y_i}) \Delta t$	Vertical displacement for horizontal launch
$v_{x_i} = \frac{x_f}{\Delta t}$	Initial horizontal velocity
$y_f = y_i + (v_{y_i}) \Delta t - \frac{1}{2}g(\Delta t)^2$	Vertical displacement for angled launch
$v_{y_i} = \frac{\frac{1}{2}g(\Delta t)^2 - y_f}{\Delta t}$	Initial vertical velocity for angled launch
$v_i = \frac{v_{y_i}}{\sin \theta}$	Total initial velocity from vertical component
$R = \frac{v_i^2 \sin 2\theta}{g}$	Range of projectile
$\theta = \frac{1}{2} \sin^{-1} \left( \frac{gR}{v_i^2} \right)$	Angle required to hit a target at distance R

Additional info:

• Gravity ($g$) is typically taken as $9.8$ m/s2.

• Accuracy in measurement of time and distance is crucial for reliable results.

• Projectile motion assumes negligible air resistance.