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The range of a projectile, such as a ski-jumper leaping off a hill, is the horizontal distance it covers before landing. Understanding the range is crucial in predicting where the jumper will land. The formula for the range \( R \) of a projectile is derived from its launch speed \( v \), launch angle \( \theta \), and the acceleration due to gravity \( g \). The basic expression for calculating the range is

• \( R = \frac{v^2}{g} \sin 2\theta \)

This equation tells us the range is dependent on the square of the initial velocity and the sine of two times the launch angle.
For the ski-jumper, the angle \( \phi \) of the hill with the horizontal also factors into the range calculation. Thus, the modified equation becomes:

• \( R = \frac{v^2}{g} \sin (2\theta - \phi) \)

This modification adjusts the projectile's range to account for the slope of the hill, giving a more accurate measure of how far the jumper travels parallel to the horizontal plane.

The launch angle \( \theta \) significantly affects the trajectory and the range of a projectile. The goal in scenarios like ski-jumping is often to achieve the maximum possible range, and specific angles result in optimal distances.
In traditional projectile motion without considering additional factors, the ideal launch angle for maximum range is typically \( 45^{\circ} \). However, when a slope like in our ski-jumping example exists, the equation adapts. As per the problem, this modified optimal launch angle is:

• \( \theta = 45^{\circ} - \frac{\phi}{2} \)

Here, \( \phi \) is the angle of the hill with respect to the horizontal.
This adjustment accounts for the slope's effect on the projectile, ensuring that the launch angle maximizes the range along the inclined surface of the hill. Understanding how to adapt the launch angle for different surfaces or slopes is essential for achieving the longest possible jump.

Differentiation is a mathematical tool that allows us to find the rate at which a function changes. In physics, it's widely used to solve problems involving optimization, like finding the maximum range of a projectile. In our ski-jumper example, to find the angle \( \theta \) that maximizes the range \( R \), we differentiate the range equation concerning \( \theta \):

• \( \frac{dR}{d\theta} = \frac{2v^2}{g} \cos(2\theta - \phi) \)

Setting the derivative equal to zero allows us to find the points where the range is at a maximum or minimum.
For the ski-jumper, we solve

• \( \cos(2\theta - \phi) = 0 \)

This equation implies that \( 2\theta - \phi = 90^{\circ} \). Solving for \( \theta \) gives us the optimal launch angle. Differentiation helps us determine the exact conditions under which a physical quantity, like the range, reaches its extremum, offering a methodical approach to optimization in physics.