91Ó°ÊÓ

Projectile motion is a common type of two-dimensional motion that we often analyze, especially in physics. In the scenario of a cannonball being fired, the motion is determined by the initial velocity, angle, and the forces acting on it, primarily gravity.
A projectile follows a parabolic path called a trajectory. This trajectory depends on the initial speed and angle of launch, which, in our example, is given by the function:\[ h(x) = x - 32 \left(\frac{x}{370}\right)^2 \]This equation models the height \( h \) of the cannonball based on the horizontal distance \( x \). The term \(-32 \left(\frac{x}{370}\right)^2\) represents the effect of gravity, pulling the projectile back down to the ground.
Understanding projectile motion helps in predicting where an object will land and how high it will go.

The derivative in mathematics speaks to how a function changes as its input changes. In physics, it often answers questions about rates of change, such as velocity and acceleration.
For our function:\[ h(x) = x - 32 \left(\frac{x}{370}\right)^2 \]the derivative \( \frac{dh}{dx} \) gives us the rate at which the height \( h \) changes as the horizontal distance \( x \) changes.
At a specific point, like 3000 feet downrange, calculating the derivative tells us the nature of the motion at that point—whether the object is moving upwards or downwards.
- A positive derivative implies the height is increasing as \( x \) increases.- A negative derivative indicates the height is decreasing, meaning the projectile is descending.
In our problem, the calculated derivative \(-0.402\) at 3000 feet suggests that the cannonball is indeed descending at that point.

Plotting functions allows us to visualize their behavior over a range of values. For instance, the function given for the motion of the cannonball can be plotted over a range of \( x \) values to visualize its trajectory. To plot:

• Select a range of \( x \) values.
• Calculate \( h(x) \) for each \( x \).
• Use graphing tools or software to display the points and connect them smoothly.

The result is a parabola, typical of projectile motion, showing the rise and fall due to the projectile's initial upward motion and subsequent pull of gravity.
Each point on the graph helps predict the height of the cannonball at a particular distance and assists in visualizing where its maximum and minimum points may lie.

Calculating the height of an object in motion at a specific point is crucial for understanding its behavior at that moment. In our cannonball example, we need the height at 3000 feet downrange.
By substituting \( x = 3000 \) into the function:\[ h(x) = x - 32 \left(\frac{x}{370}\right)^2 \]we accomplish this task:

• Calculate \( \frac{3000}{370} \approx 8.1081 \).
• Square the result: \( (8.1081)^2 \approx 65.73 \).
• Multiply by 32: \( 32 \times 65.73 \approx 2103.36 \).
• Subtract from 3000: \( h(3000) = 3000 - 2103.36 = 896.64 \)

Thus, the height of the cannonball at 3000 feet is approximately 896.64 feet. This value not only tells us how far above the ground the cannonball is but also allows us to understand its path visually when plotted.