Physics Lab Report: Ball Position Prediction

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Introduction

A toy company is now making an instructional videotape on how to predict the position. Therefore, in order to make the prediction accurate, how the horizontal and vertical components of a ball’s position as it flies through the air should be understood. This experiment is to calculate functions to represent the horizontal and vertical positions of a ball. It does so by measuring and calculating the components of the position and velocity of the ball during the toss. Therefore, we can also calculate the acceleration during the procedure.

Prediction

We establish a coordinate system with the x-axis representing the horizontal direction and the y-axis representing the vertical direction.

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Considering that the only force acting on the ball during the toss is gravity, we anticipate the following:

The ball should maintain a constant horizontal speed (the horizontal component of initial velocity).
Vertically, gravity will act downward, resulting in a constant acceleration of -9.8 m/s².
The ball will initially move upward due to its vertical velocity and then begin to descend.
The position graph will form a parabolic shape.

The mathematical representation of the ball's motion can be expressed as follows:

x(t) = x₀ + (V_ox * t)
y(t) = y₀ + (V_oy * t) - (0.5 * g * t²)

Here, V_ox and V_oy are the initial horizontal and vertical velocities, and x₀ and y₀ are the initial horizontal and vertical positions.

Procedure

A 146.12g spherical ball is thrown upward obliquely.

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Its toss trajectories were recorded by a video camera. The Motion Lab analysis software was used to generate the graph of the position and velocity as functions of time both horizontally and vertically.

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The horizontal position and velocity versus time function were fit by eye as oblique line and horizontal line. The vertical x(t) and v(t) functions were parabola and oblique line. The acceleration should be the slope of the velocity of both components.

Data Analysis

Our analysis of the data yielded the following results:

The horizontal velocity (V_x) versus time graph remained a horizontal line, indicating a constant velocity of -1.5 m/s.
The horizontal position (x(t)) as a function of time followed a linear relationship: x(t) = 0.4m - (1.5m/s) * t
The vertical velocity (V_y) versus time graph was proportional to time and started positive, then decreased to zero when the ball reached its highest point.
The vertical position (y(t)) as a function of time was a parabolic function: y(t) = 0.5m + (2.8m/s) * t - (4.9m/s²) * t²

From the data, we calculated the following parameters:

Horizontal acceleration (a_x): 0 m/s²
Vertical acceleration (a_y): -9.8 m/s²
Initial horizontal velocity (V_ox): -1.5 m/s
Initial vertical velocity (V_oy): 2.8 m/s
Initial horizontal position (x₀): 0.4 m
Initial vertical position (y₀): 0.5 m

Conclusion

The result of the experiment is generally the same with the prediction. However there is still some factors that lead to some minor inaccuracy. One factor is the distortion of the camera. Since the camera is recording from the center-most portion of the field, the graph it is plotting may not be so accurate. Another one is the air resistance. During the toss, the ball experience the air resistance to some extent both vertically and horizontally. However when we are doing the prediction, the resistance is neglected, But it does exist, though minor, in the actual experiment. In the procedure, we get -0.2m/s as the vertical velocity at the highest point. However as we predicted, it should be 0m/s. This is probably because the point we are choosing is not actually the real highest point. The camera is not precise enough to record any single change of the ball.

Generally the horizontal velocity does not change and the position is promotional to time. But the vertical velocity decreases and the vertical position increases and then decreases.The ball have the maxim velocity when it hit the ground and have the smallest at the highest point.