Problem 35 - Calculus of a Single Variable

Question

Projectile Motion The range \(R\) of a projectile is $$R=\frac{v_{0}^{2}}{32}(\sin 2 \theta)$$ where \(v_{0}\) is the initial velocity in feet per second and \(\theta\) is the angle of elevation. Use differentials to approximate the change in the range when \(v_{0}=2500\) feet per second and \(\theta\) is changed from \(10^{\circ}\) to \(11^{\circ} .\)

Step-by-Step Solution

Answer

The approximate range change when the angle of elevation changes from \(10^{\circ}\) to \(11^{\circ}\) with an initial velocity of 2500 ft/s is the value obtained from evaluating the expression \(\Delta R\).

Step 1: Understand the given formula and variables

The given formula is \(R=\frac{v_{0}^{2}}{32}(\sin 2 \theta)\), where \(R\) represents the range of the projectile, \(v_{0}\) is the initial velocity, and \(\theta\) is the angle of elevation. In this problem, \(v_{0}=2500\) feet per second and \(\theta\) is changed from \(10^{\circ}\) to \(11^{\circ}\). The main task is to compute the change in the range, \(\Delta R\), due to this change.

Step 2: Express the variables in terms of radians

The trigonometric function in the formula, \(\sin\), operates on angles expressed in radians. Thus, \(\theta\) should be converted from degrees to radians. 1 degree is equal to \(\frac{\pi}{180}\) radians. Therefore, \(10^{\circ} = 10 * \frac{\pi}{180}\) radians and \(11^{\circ} = 11 * \frac{\pi}{180}\) radians.

Step 3: Find the differential of \(R\)

The differential, \(dR\), is found by taking the derivative of \(R\) with respect to \(\theta\) and then multiplying by \(d\theta\). Using the chain rule,\(dR = \frac{dR}{d\theta} * d\theta = \frac{v_{0}^{2}}{16} * \cos(2\theta) * d\theta\).

Step 4: Calculate the change in \(R\)

To calculate the change in \(R\) with \(\theta\) changing from \(10^{\circ}\) to \(11^{\circ}\), use the formula for \(dR\), replace \(d\theta\) with the change in \(\theta\) which is \(1 * \frac{\pi}{180}\) radians, and use the initial value of \(\theta = 10 * \frac{\pi}{180}\) for the cosine function. \(\Delta R = dR = \frac{v_{0}^{2}}{16} * \cos(2\theta) * d\theta = \frac{2500^2}{16} * \cos(2*10 * \frac{\pi}{180}) * (1 * \frac{\pi}{180})\) feet.

Key Concepts

Differentials in Calculus

Differentials in calculus are tools that enable us to approximate tiny changes in functions given small changes in their inputs. This concept is highly utilitarian in evaluating how sensitive a function is to variations in its arguments. For instance, if we consider the projectile motion problem, the range, denoted as \(R\), is a function of the angle of inclination, \(\theta\). When there is a small change in the angle, the differential \(dR\) tells us approximately how much the range \(R\) will change.

The calculation of \(dR\) involves taking the derivative of \(R\) with respect to \(\theta\) and then multiplying by an infinitesimally small change in angle, \(d\theta\). It's a two-step approach: first compute the derivative (which gives us the rate of change of \(R\) at a specific angle), and then estimate the actual change by using the differential. This method is excellent for finding approximate changes without the need for exact and often complex calculations.

Trigonometric Functions

Trigonometric functions are a cornerstone in calculus, especially when dealing with periodic phenomena or problems involving triangles. These include sine (sin), cosine (cos), tangent (tan), and their reciprocals. In projectile motion, the sine function is used to determine the range \(R\).

The formula \(R = \frac{v_{0}^{2}}{32}(\sin 2 \theta)\) showcases how the range is directly affected by the sine of twice the angle of elevation. These functions take an angle as an input and return a ratio of sides in a right-angle triangle. For example, the sine of an angle in a right-angle triangle is the ratio of the opposite side to the hypotenuse. In physics, we often use these functions to translate rotational motion into linear motion, just as in the case of projectile motion.

Derivative Applications

The derivative is a fundamental concept in calculus that represents the rate at which a function is changing at any given point. Practically, it allows us to predict how a system behaves under small changes in its variables. In the context of projectile motion, we apply the derivative to predict the change in the range of a projectile with respect to changes in the angle of elevation using the provided formula.

The application of derivatives, outlined in Step 3 of the given solution, involves computing the rate of change of the projectile's range by differentiating the range equation with respect to \(\theta\). This step is crucial for understanding how a tiny change in the launch angle could potentially affect the distance traveled by the projectile.

Angle Conversion Radians Degrees

Angles can be measured in degrees or radians. In calculus, we often convert angles from degrees to radians because the trigonometric functions in the calculus context require angles in radians. The conversion factor between these two units is \(\frac{\theta}{180} = \frac{\theta^{\circ}}{\pi}\), where \(\theta^{\circ}\) is the angle in degrees and \(\theta\) is the angle in radians.

The mentioned projectile motion problem demonstrates this conversion: \(10^{\circ}\) and \(11^{\circ}\) are translated into radians before applying the calculus methods. The conversions, as shown in Step 2, form the base for accurate differential calculations, allowing students to see the direct relation between angular adjustments and their quantitative impacts on projectile range.