Problem 40 - Single Variable Calculus

Question

A projectile is fired from a cannon that makes an angle of \(\theta\) degrees with the horizontal. If the muzzle velocity is constant, then the range in feet of the projectile is a function of \(\theta\), that is, \(R=f(\theta)\). a. What is the physical meaning of \(f^{\prime}(\theta) ?\) Give units. b. What can you say about the sign of \(f^{\prime}(\theta)\), where \(0^{\circ}<\theta<90^{\circ} ?\) c. Given that \(f(40)=10,000\) and \(f^{\prime}(40)=20\), estimate the range of a projectile if it is fired at an angle of elevation of \(41^{\circ}\).

Step-by-Step Solution

Answer

(a) The physical meaning of \(f^{\prime}(\theta)\) is the rate of change of the projectile range with respect to the launch angle \(\theta\), with units \(ft/degree\). (b) The sign of \(f^{\prime}(\theta)\) is positive for \(0^\circ < \theta < 45^\circ\) and negative for \(45^\circ < \theta < 90^\circ\). (c) The range of a projectile fired at an angle of elevation of \(41^\circ\) is approximately \(10,020\) feet.

Step 1: (a. Find the physical meaning of \(f^{\prime}(\theta)\))

The range of the projectile, denoted as \(R=f(\theta)\), is a function of the launch angle \(\theta\). The physical meaning of its derivative, \(f^{\prime}(\theta)\), can be understood as the rate of change of the projectile range with respect to the launch angle \(\theta\). In other words, it indicates how the range of the projectile will change as the launch angle changes. The units of this value will be the ratio between the units of the range (\(ft\)) and the units of the launch angle (degrees), so it will be \(ft/degree\).

Step 2: (b. Determine the sign of \(f^{\prime}(\theta)\))

To determine the sign of the derivative \(f^{\prime}(\theta)\) for \(0^\circ < \theta < 90^\circ\), we need to consider the general behaviour of the range function as a function of the angle. We know that the range will be zero when launched vertically upwards (\(\theta = 90^\circ\)) or horizontally (\(\theta = 0^\circ\)). Between these two points, there is an angle for which the range is maximum, which happens at \(\theta = 45^\circ\). Thus, the derivative \(f^{\prime}(\theta)\) will be positive for \(0^\circ < \theta < 45^\circ\), meaning that the range of the projectile is increasing as the launch angle increases, and it will be negative for \(45^\circ < \theta < 90^\circ\), meaning that the range is decreasing as the launch angle increases.

Step 3: (c. Estimate the range of the projectile at an angle of \(41^\circ\))

We are given that \(f(40) = 10,000\) and \(f^{\prime}(40) = 20\). We are asked to estimate the range of the projectile if it is fired at an angle of elevation of \(41^\circ\). Since \(41^\circ\) is close to \(40^\circ\) and we have information about \(f(40)\) and \(f^{\prime}(40)\), we can use a linear approximation to estimate the range at \(41^\circ\). Using the tangent line to the curve \(R=f(\theta)\) at \(\theta=40^\circ\), we can approximate the range as follows: \(R(41) \approx R(40)+ f^{\prime}(40)(41-40)\) \(R(41) \approx 10000 + 20(41-40)\) \(R(41) \approx 10000 + 20(1)\) \(R(41) \approx 10,020\) Therefore, the range of a projectile fired at an angle of elevation of \(41^\circ\) is approximately \(10,020\) feet.

Key Concepts

Rate of Change

Understanding the rate of change is crucial when studying the dynamics of a projectile's motion. In the given exercise, the rate of change is represented by the function's derivative, denoted by \( f'(\theta) \). This derivative measures how sensitive the projectile's range (the horizontal distance the projectile travels) is to changes in the launch angle \( \theta \).

For instance, a positive value of \( f'(\theta) \) indicates that a small increase in the launch angle will result in an increase in the projectile's range. Conversely, a negative value suggests that increasing the angle will decrease the range. This concept enables us to predict the behavior of the projectile's path and optimize for desired outcomes such as maximum distance.

Derivative Application

The application of derivatives is a powerful tool in calculus, particularly for analyzing motion. In projectile motion, the derivative of the range function with respect to the angle \( f'(\theta) \) has a clear physical interpretation. It is the rate at which the range changes per degree change in the angle.

Example in Projectile Motion

If we know that \( f'(40) = 20 \) ft/degree, this means that at a 40-degree launch angle, for each additional degree we tilt the cannon upward, the projectile will, on average, travel an extra 20 feet. It is important to note that this rate changes as the angle changes, demonstrating the non-linear relationship between the angle of launch and the range of a projectile.

Linear Approximation

Linear approximation is a technique used in calculus to estimate the values of a function at a certain point using the value and slope of the function at a nearby point. The underlying assumption is that within a small interval, the function behaves approximately linearly.

In our projectile motion scenario, we use linear approximation to estimate the range \( R(41) \) based on the known range and rate of change at 40 degrees. By assuming the function is linear between 40 and 41 degrees, we use the formula:

• \( R(41) \approx R(40) + f'(40)\times (41 - 40) \)

This method allows us to quickly estimate outcomes without the need for complex calculations, and it is particularly useful when dealing with functions that are difficult to express explicitly.

Optimizing Projectile Range

Optimizing the projectile range involves finding the launch angle \( \theta \) that maximizes the horizontal distance traveled by the projectile. This is a key concept in problems dealing with projectile motion and is closely tied to the application of derivatives.

As the angle increases from 0 to 45 degrees, the derivative \( f'(\theta) \) will be positive, indicating increasing range. At 45 degrees, the range reaches a maximum and \( f'(\theta) \) should be 0, as the instantaneous rate of change at this peak is zero. Any angle beyond this will result in a negative derivative, signifying a decrease in range. By utilizing calculus to find where the derivative of the range function equals zero, we can determine the optimal launch angle for maximizing the projectile's distance.