Problem 55 - Essential Calculus

Question

The path of a projectile is modeled by the parametric equations $$ x=\left(90 \cos 30^{\circ}\right) t \quad \text { and } \quad y=\left(90 \sin 30^{\circ}\right) t-16 t^{2} $$ where \(x\) and \(y\) are measured in feet. (a) Use a graphing utility to graph the path of the projectile. (b) Use a graphing utility to approximate the range of the projectile. (c) Use the integration capabilities of a graphing utility to approximate the arc length of the path. Compare this result with the range of the projectile.

Step-by-Step Solution

Answer

As the arc length and the range both give measures of the distance covered by the projectile, you will find that the arc length is greater than the range. This reflects the fact that the path of the projectile is not a straight line, and hence covering this path involves traveling a greater distance than simply the straight-line distance between its start and end points.

Step 1: Graphing the Path

Use a graphing utility to graph the equations \(x=(90 \cos 30^\circ) t\) and \(y=(90 \sin 30^\circ) t - 16t^2\). This will give you a visual representation of the path of the projectile.

Step 2: Approximate the Range

The range of the projectile is the maximum 'x' value that the projectile reaches. This occurs when the 'y' value is zero. So substitute \(y=0\) in the equation \(y=(90 \sin 30^\circ) t - 16t^2\) to solve for 't' and substitute this 't' in the 'x' equation to get the range.

Step 3: Approximate the Arc Length

The arc length of the path is given by the formula \(\int_{a}^{b} \sqrt{[dx/dt]^{2} + [dy/dt]^{2}} dt\) where 'a' and 'b' are the initial and final time respectively. Calculate the derivatives of 'x' and 'y' with respect to 't', substitute them in the formula and perform the integration to get the approximate arc length. As you can see, these calculations are more complex than that of the range, reflecting the fact that the path of the object is not a straight line.

Key Concepts

Projectile Motion

Projectile motion is a fascinating topic where physics and mathematics come together to describe the path of an object that is thrown into the air. In this scenario, the object moves in a curved trajectory under the influence of gravity. The parametric equations governing the projectile's motion are

• \( x = (90 \cos 30^{\circ}) t \)
• \( y = (90 \sin 30^{\circ}) t - 16t^2 \)

Parametric equations are useful because they allow us to track the movement in both horizontal and vertical directions simultaneously. The angle \(30^{\circ}\) tells us the direction in which the projectile was launched. The speed multiplier \(90\) gives the initial speed at which the projectile was launched. The term \(-16t^2\) in the y-equation accounts for the acceleration due to gravity pulling the projectile downward as it flies through the air.

Graphing Utility

Graphing utilities are essential tools in mathematics that help us visualize complex equations and data. To graph the projectile's path, we can use a graphing calculator or software. These tools provide a visual representation of the trajectory, making it easier to understand the motion.

When using a graphing utility for the parametric equations given, you will see the projectile follows a curved path known in mathematics as a parabola. This shape is typical for objects under projectile motion without any air resistance. Graphing utilities can also be used to approximate certain values, such as the range of the projectile. By observing where the trajectory crosses the x-axis, we can determine how far the projectile travels from the launch point.

Integration and Differentiation

Integration and differentiation are fundamental concepts in calculus, crucial for analyzing projectile motion thoroughly. Differentiation helps us find rates of change; for example, the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) give us the horizontal and vertical velocities of the projectile.

• To understand the path length – the arc length – we utilize integration. This involves the formula: \(\int_{a}^{b} \sqrt{[dx/dt]^{2} + [dy/dt]^{2}} dt\).

This formula calculates the continuous sum of small straight segments along the projectile's path to find the total path length it travels, which is different from just the range, since the path is not a straight line.

In practical terms, these calculus operations might be performed using a graphing utility with integration capabilities. This approach eases the complexity, allowing users to input the functions and receive solutions without performing manual calculations, simplifying tasks for students and professionals alike.