Projectile Motion Feedback Pre Lab 6 Doesn't Answer The Ques ✓ Solved

Projectile Motionfeedbackpre Lab 6 Doesn't Answer The Questioni Don

The assignment requires analyzing projectile motion through experimental and theoretical approaches, including calculations of initial velocities, angles, and the effects of air resistance. It involves performing multiple tests with varied parameters such as launch angle, initial height, and presence or absence of air resistance, and comparing the measured ranges with predictions based on kinematic equations. The task also includes evaluating the impact of air resistance, estimating the minimum launch speed for a record baseball throw, and discussing uncertainties encountered in measurements.

Sample Paper For Above instruction

Introduction

Projectile motion is a fundamental concept in physics, describing the motion of an object launched into the air under the influence of gravity, neglecting air resistance. It is essential to understand how different parameters—such as launch angle, initial velocity, initial height, object mass, and diameter—affect the range of the projectile. This experiment aims to verify the applicability of two-dimensional kinematic equations to real-world projectile motion and to analyze the effects of air resistance under various conditions.

Methodology

The experiment was conducted by launching projectiles from a cannon at different angles and initial heights. The key parameters manipulated included launch angle, initial height, initial velocity, object mass, diameter, and air resistance coefficient. Both with and without air resistance, multiple trials were performed, measuring the horizontal and vertical distances to compare with theoretical predictions.

Calculations of Initial Velocity

A pivotal aspect was determining the initial velocity needed to hit a target at known distance, using kinematic equations. Assuming negligible air resistance, the horizontal motion obeys the equation \(\Delta x = v_{ox} t\), where \(\Delta x\) is the horizontal displacement, \(v_{ox}\) the horizontal component of the initial velocity, and \(t\) the time of flight. The vertical motion is governed by \(\Delta y = v_{oy} t + \frac{1}{2} a_y t^2\), with \(a_y = -g\). Combining these equations allows solving for initial velocity and launch angle for a desired range.

For example, to hit a target 24 meters away with the cannon at the same height, the calculation involved solving:

\[

\Delta y = 0 \Rightarrow v_{oy} = v_0 \sin \theta

\]

\[

\Delta x = v_{ox} t \Rightarrow t = \frac{\Delta x}{v_0 \cos \theta}

\]

and the vertical motion equation:

\[

0 = v_{oy} t - \frac{1}{2} g t^2

\]

which, combined, yields a relation between \(v_0\) and \(\theta\).

In the pre-lab calculations, initial velocity was estimated as approximately 21.68 m/s at an angle of roughly 15°, which was used in predicting the projectile’s landing point, although the actual impact point varied slightly due to simplifications.

Effect of Parameters on Range

In the absence of air resistance, the range is maximized at a launch angle of 45°, consistent with theoretical analysis:

\[

R = \frac{v_0^2 \sin 2\theta}{g}

\]

where \(R\) is the range.

Increasing the initial height tends to increase the range because the projectile has more time to travel horizontally before impact. Likewise, increasing initial velocity increases range, as predicted by the quadratic relation.

Mass and diameter, under ideal conditions without air resistance, do not influence the range; this is supported by the principle of independence due to inertia, where air resistance is negligible.

In the presence of air resistance, the dynamics become more complex:

- Increasing the angle from zero increases the range up to about 45°, after which it decreases.

- Increasing initial speed generally increases the range but with diminishing returns due to drag.

- Increasing diameter tends to decrease the range because larger objects experience more air resistance.

- Increasing mass, surprisingly, does not significantly affect the range if drag coefficient is constant, aligning with the understanding that air resistance depends primarily on size and shape.

Air Resistance Effects

Experimental tests with air resistance turned on and off demonstrated that air resistance significantly affects the range, especially at higher velocities and larger diameters. When air resistance is negligible, the measured range closely matches the theoretical value derived from the kinematic equations. Conversely, with air resistance, the range is often reduced, especially at higher initial speeds.

Air resistance becomes negligible when the object has a small diameter, low velocity, or when the projectile is launched at angles close to 45° where the trajectory duration is optimized, and drag effects are proportionally reduced.

Predicting Projectile Impact with Known Parameters

Using the equations for projectile motion, a prediction for hitting a target 24 meters away was calculated with:

\[

v_0 = 21.68\, \text{m/s}

\]

\[

\theta \approx 15^\circ

\]

which predicted a landing distance of about 28 meters, slightly overshooting the target. Slight adjustments to the initial velocity and angle, refined through trial, are often necessary for precise hitting.

Maximum Launch Distance at Elevated Heights

With the cannon significantly above ground level, the angle for maximum horizontal distance shifts away from the classic 45°, as observed in the simulated results. At an initial velocity of 10 m/s and initial height of 5 meters, the experimental optimal angle was found around 35°, close to theoretical expectations that increasing initial height can extend the projectile’s range at lower launch angles.

Air Resistance in Record Baseball Throw

Simulating the world record baseball throw with air resistance included, the minimum initial speed required to achieve a 445-foot (approximately 135.6 meters) distance was estimated around 70 m/s (about 156.6 mph), with an optimal launch angle approximately 45°. The initial height was estimated at 5 meters, considering release points from a pitcher’s hand. These estimates were obtained through iterative simulation and guess-and-check methods, acknowledging that real-world factors like tailwind and ball spin are not simulated.

Uncertainties and Experimental Limitations

Measurements of distances were affected by the wear and curvature of the tape measure, uneven ground, and measurement errors within approximately 1 mm. Variability in measurements was accounted for with multiple trials, and uncertainties were estimated accordingly.

Conclusion

The experiment confirmed that the primary parameters influencing projectile range, in ideal conditions, are the launch angle and initial velocity, consistent with theoretical equations. Air resistance plays a significant role for higher speeds and larger objects, reducing the range. The predictions for hitting specific targets with calculated initial velocities were close but required adjustments due to real-world uncertainties and simplified assumptions in the calculations.

Future improvements could include more accurate measurements of initial velocities using high-speed cameras, and controlling surface levelness to minimize measurement errors. The theoretical models align well with experimental data under controlled conditions, demonstrating the robustness of the two-dimensional kinematic equations when supplemented with empirical trial adjustments.

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