Question

A steel ball will drop \(16 t^{2}\) feet in \(t\) seconds. Such a ball is dropped from a height of 64 feet at a horizontal distance 10 feet from a 48-foot street light. How fast is the ball's shadow moving when the ball hits the ground?

Short answer

The shadow moves at a rate of 32 feet/second when the ball hits the ground.

Step-by-step solution

Understand the Problem

We need to find out how fast the shadow of a ball is moving when the ball hits the ground. The ball is dropped from 64 feet, 10 feet away from a streetlight that is 48 feet tall.

Determine Time to Hit Ground

We know the ball is dropped from 64 feet, and it follows the motion equation: \( h(t) = 64 - 16t^2 \). Solve for \( t \) when \( h(t) = 0 \) to find the time it takes to hit the ground.\[ 64 - 16t^2 = 0 \]\[ 16t^2 = 64 \]\[ t^2 = 4 \]\[ t = 2 \]}},{

Key concepts

Motion Equations

In this problem, motion equations are used to describe the vertical movement of the steel ball as it falls to the ground. The motion equation given is \( h(t) = 64 - 16t^2 \), where \( h(t) \) represents the height of the ball at any time \( t \), and \( t \) is measured in seconds.
This particular equation is derived from the basic physics of gravity, where an object falls at a constant acceleration, here simplified to 16 ft/sec² due to the square term. The negative sign indicates that as time progresses, the height decreases.
Understanding motion equations is crucial because they allow us to predict the position of a moving object at any point in time. In this scenario, solving \( 64 - 16t^2 = 0 \) helps us find that the ball hits the ground at \( t = 2 \) seconds. This foundational step is paramount before addressing how fast the shadow is moving. Recognizing the structure of motion equations enables you to model various real-world physics problems accurately.

Calculus Problem Solving

Calculus offers powerful tools for analyzing changes, which is precisely what related rates problems entail. Here, we're interested in how quickly the ball's shadow moves. This involves understanding the rate of change of various quantities.
In the problem given, the setup involves differentiating the given relationship among distances with respect to time. Since calculus allows us to analyze how a particular quantity changes, it helps us find the velocity of the shadow. For such problems, implicit differentiation is often employed. By relating the rates of change, we determine the speed of the shadow.
A typical related rates problem involves:

• Finding the known quantities and the relationships between them.
• Taking the derivative with respect to time using implicit differentiation.
• Substituting the known values to find the unknown rate.

In this context, calculus problem solving transforms the problem of a moving shadow into a manageable mathematical model.

Physics Applications

Applying physics concepts can solve everyday problems that involve motion, forces, and energy. In this exercise, the main physics application is analyzing how a shadow moves in conjunction with a falling object.
The relationship between the physical dimensions involves geometric reasoning, where the heights create similar triangles with the distances involved. The use of light sources and shadows illustrates principles of optics and dynamics.
By applying these physics concepts correctly, you understand that the motion equations governing the ball also pertain to the shadow's movement. You realize that since the streetlight does not move, the shadow's movement entirely depends on the ball's motion. This insight aligns with principles such as the conservation laws and dynamic constraints of the system.
Understanding these physical applications not only helps solve theoretical problems but also equips us with analytical skills critical for solving practical, real-life situations.