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Chapter 13: Problem 12

The parametric equations for the paths of two projectiles are given. At what rate is the distance between the two objects changing at the given value of \(t\) ? $$ \begin{aligned} &x_{1}=48 \sqrt{2} t, y_{1}=48 \sqrt{2} t-16 t^{2} \\ &x_{2}=48 \sqrt{3} t, y_{2}=48 t-16 t^{2} \\ &t=1 \end{aligned} $$

Short Answer

Expert verified
Evaluating this expression gives us the rate at which the distance between the two objects is changing at \(t = 1\).

Step by step solution

01

Calculate the distance between the two objects (Distance Formula)

We can use the distance formula between two points in a plane to calculate the distance between the two objects. The distance \(d\) is given by: \[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} = \sqrt{(48 \sqrt{3} t - 48 \sqrt{2} t)^2 + ((48t - 16t^2) - (48 \sqrt{2} t - 16 t^2))^2}\]
02

Simplify the expression

Next, we should simplify the above expression to get a function in terms of \(t\) for the distance \(d(t)\).Simplifying, we get:\[d(t) = \sqrt{768t^2 - 6912t^4 + 1024t^4}\]
03

Differentiate \(d(t)\)

The rate at which the distance between the two objects is changing at \(t = 1\) is given by the derivative of \(d(t)\) at \(t = 1\). Let's find that derivative using chain rule and power rule:\[d'(t) = \frac{1}{2} (768t^2 - 6912t^4 + 1024t^4)^{-\frac{1}{2}} (2*768t - 4 * 6912t^3 + 4 * 1024t^3)\]
04

Evaluate the derivative at \(t = 1\)

Now we substitute \(t = 1\) into \(d'(t)\) to find the rate of change of distance at that specific time. \[d'(1) = \frac{1} {2} (768 - 6912 + 1024)^{-\frac{1}{2}} (768 - 4 * 6912 + 4 * 1024)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
The distance formula is essential when we want to find the distance between two points in a Euclidean plane. It's derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For any two points with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \) , the distance between them, denoted as \( d \), is given by this formula:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
This equation is vital for many applications in geometry, physics, and other areas. For instance, when calculating the distance a projectile has traveled or finding the shortest path between two locations on a map.
Derivative of Distance
When we calculate the derivative of a distance function with respect to time, we're measuring the rate at which that distance is changing over time. It's a foundational concept in the study of motion and is especially important in physics and engineering.

In the context of parametric equations, where \( x \) and \( y \) are both functions of a third parametric variable \( t \), often time, we find the rate of change in distance at a specific time by differentiating the distance function \( d(t) \) with respect to \( t \) using the chain rule and power rule. This process reveals the velocity or speed at which the two points are moving apart or coming together.
Chain Rule
The chain rule is a formula to compute the derivative of a composition of two or more functions. In simple terms, if you have a function \( h(x) = g(f(x)) \), and you want to find \( h'(x) \)鈥攖he rate at which \( h \) changes with respect to \( x \)鈥攜ou would multiply the derivative of \( g \) with respect to \( f \) by the derivative of \( f \) with respect to \( x \):
\[ h'(x) = g'(f(x)) \cdot f'(x) \]
The chain rule is widely used when dealing with derivatives of nested functions, like the distance function, which relies on the square root of squares of differences of parametric functions.
Power Rule
The power rule is one of the most basic and widespread rules in differentiation. It is used when we need to find the derivative of a function that is a monomial, which is a single term consisting of a variable raised to a power, like \( x^n \). The rule is elegantly simple:

If \( f(x) = x^n \) for some real number \( n \) , then its derivative \( f'(x) \) is:
\[ f'(x) = n \cdot x^{(n-1)} \]
The power rule makes finding derivatives straightforward and is particularly handy with polynomial functions. In our exercise, using the power rule helps to differentiate terms like \( t^2 \) and \( t^4 \) efficiently.

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Most popular questions from this chapter

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