Laboratory Report: Projectile Motion

Abstract

The purpose of this experiment is to launch a metal ball from the launcher at different angles using basic knowledge of projectile motion. Based on the equations of projectile motion, it can be concluded that the movement of the projectile depends on the initial velocity of the projectile, the angle of launch, and the acceleration due to gravity. This experiment aims to provide a practical representation of projectile motion, enabling us to conduct experiments by recording data, performing calculations, and drawing conclusions about the projectile's motion equation.

Introduction

The concept of projectile motion describes the motion of an object, such as a missile or a metal ball, that is launched at a certain angle and velocity. Once launched, the projectile follows a specific trajectory, which can be represented graphically. It is essential to note that the only force acting on the projectile during its flight is the acceleration due to gravity. However, this acceleration is often neglected in problem-solving, which can lead to inaccuracies in calculations.

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The objective of this laboratory work is to conduct ten experiments and analyze the obtained theoretical and practical values using provided equipment.

Experimental Data

Table 1: Recorded Data

Data Analysis

Theory

In order to find the vertical component (y-component) of the projectile's velocity as a function of time, we use the following equation:

$$V_{y} = V_{0} cdot sin(θ) - g cdot t$$

Where:

• $$V_{y}$$ - Vertical component of velocity
• $$V_{0}$$ - Initial velocity of the projectile
• $$g$$ - Acceleration due to gravity
• $$t$$ - Time
• $$V_{0y}$$ - Initial vertical component of velocity
• $$θ$$ - Angle of launch

At the peak of the trajectory, the vertical component of the projectile's velocity is zero.

Therefore:

$$0 = V_{0} cdot sin(θ) - g cdot t_{p}$$

Where:

• $$t_{p}$$ - Time spent to reach the peak of the trajectory

According to the equation above, the total time of flight is twice the time to reach the peak:

$$T = 2 cdot t_{p}$$

Substituting the total time of flight into the equation for horizontal distance, we can derive the formula for horizontal range ($$D$$) as a function of initial velocity and launch angle:

$$D = frac{V_{0}^{2} cdot sin(2θ)}{g}$$

Where:

• $$D$$ - Horizontal range of flight
• $$V_{0x}$$ - Initial horizontal component of velocity

Calculation of Initial Velocity

The initial velocity for each angle can be calculated using the formula for horizontal range. For example, for an angle of 25° and a range of 0.727 m:

$$0.727 = frac{V_{0}^{2} cdot sin(2 cdot 25°)}{g}$$

Solving for $$V_{0}$$:

$$V_{0} = sqrt{frac{0.727 cdot g}{sin(2 cdot 25°)}}$$

Using the standard value for acceleration due to gravity ($$g = 9.81 , text{m/s}^2$$), we can calculate the initial velocity for each angle.

Error Analysis

The error in the initial velocity for each angle can be calculated using the average error in distance and the error in angle. The distance measurement has two sources of error: the statistical error of the range measurement (standard deviation of all 10 recorded distances) and systematic error (0.001 m). These errors can be combined to calculate the final error in distance. The combination of this error with the error in angle (0.5°) yields the error in initial velocity. The results are presented in Tables 2 and 3.

For example, calculating the error for the angle of 25°:

Standard Deviation of Distance (stdev): $$0.00355 , text{m}$$

Systematic Error in Distance: $$0.001 , text{m}$$

Error in Angle: $$0.5°$$

Combining errors in distance and angle:

$$Delta D = sqrt{(0.00355 , text{m})^2 + (0.001 , text{m})^2} = 0.0037 , text{m}$$

$$Delta θ = 0.5° = 0.0087 , text{rad}$$

Calculating the error in initial velocity:

$$Delta V_{0} = V_{0} cdot sqrt{left(frac{Delta D}{D}right)^2 + left(frac{Delta θ}{θ}right)^2}$$

Using the calculated values for $$ΔD$$, $$Δθ$$, and the initial velocity ($$V_{0} = 3.07 , text{m/s}$$), we obtain the error in initial velocity for the angle of 25°.

Table 2: Calculated Distances with Errors

Table 3: Calculated Initial Velocities for Each Angle with Errors

Range vs. Angle

The general trend observed is that the range increases when the launch angle changes from 0 to 45 degrees and decreases when the angle changes from 45 to 90 degrees, as shown in Table 4. Notably, the measured distance for the angle of 35 degrees deviates significantly from the other recorded data. Several factors contribute to this deviation, including variations in the force applied to the rope (random error) and slight movements of the person performing the experiment (random error). Additionally, some deviations may occur due to difficulties in ensuring the surface provides the correct angle for launching the ball.

Table 4: Range vs. Angle

Conclusion

In conclusion, this experiment aimed to plan and conduct experiments to study the motion of objects launched at an angle to the horizontal. The goal was successfully achieved by calculating the average velocity for each angle and identifying data deviations. These deviations are essential as they provide insights into areas that require improvement to minimize errors in future experiments. The motion of an object launched at an angle to the horizontal follows a parabolic trajectory, and real-world conditions, such as air resistance, can significantly affect the projectile's range. This experiment allowed us to observe the effects of such conditions on the motion of the ball.