Modify program ba 11 from Example 4.8 to read in the...

Modify program ba 11 from Example $4.8$ to read in the acceleration due to gravity at a particular location and to calculate the maximum range of the ball for that acceleration. After modifying the program, run it with accelerations of $-9.8 \mathrm{~m} / \mathrm{sec}^{2},-9.7 \mathrm{~m} / \mathrm{sec}^{2}$, and $-9.6 \mathrm{~m} / \mathrm{sec}^{2}$. What effect does the reduction in gravitational attraction have on the range of the ball? What effect does the reduction in gravitational attraction have on the best angle $\theta$ at which to throw the ball?

Key Concepts

Projectile Motion

Projectile motion involves the motion of an object thrown into the air, subject only to the gross effects of gravity. It decomposes naturally into horizontal and vertical components, with gravity only affecting the vertical motion. Understanding this principle is crucial when analyzing the trajectory of any projectile, such as a ball.

Gravitational Acceleration

Gravitational acceleration is the rate at which an object accelerates towards the ground due to gravity. It directly influences the time a projectile remains in the air and consequently its horizontal range. A smaller magnitude of gravitational acceleration means the projectile will be in the air longer, leading to a greater horizontal distance traveled.

Projectile Range Calculation

The range of a projectile on level ground is given by the formula R = (v² sin(2?)) / |g|, where v is the initial velocity, ? is the launch angle, and g is the acceleration due to gravity. This equation demonstrates how the range increases as the magnitude of gravitational acceleration decreases, assuming initial velocity and launch angle remain constant.

Optimal Angle of Projection

The optimal angle for achieving maximum range in ideal projectile motion (neglecting air resistance) is approximately 45°. This angle maximizes the product of the horizontal and vertical components of the initial velocity. While variations in gravitational acceleration influence the overall range, the optimal launch angle remains largely unchanged under ideal conditions.

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