Question

In Exercises \(1 - 4 ,\) an object dropped from rest from the top of a tall building falls \(y = 16 t ^ { 2 }\) feet in the first \(t\) seconds. Find the average speed during the first 3 seconds of fall.

Short answer

The average speed of the object during the first 3 seconds of fall is 48 feet per second.

Step-by-step solution

Determine total distance fallen

To find the total distance fallen after 3 seconds, we substitute the time into the formula for distance given in the problem. The equation is \(y = 16t^2\), where \(t\) is the time in seconds. Substituting \(t = 3\) seconds into the equation, we obtain: \(y = 16 \cdot 3^2\).

Calculate total distance fallen

Doing the calculation above, we get \(y = 16 \cdot 9 = 144\) feet, which represents the total distance fallen after 3 seconds.

Calculate average speed

By definition, the average speed is the total distance travelled divided by the total time taken. Therefore, we divide the total distance 144 feet by the total time 3 seconds to get the average speed: \(\frac{144}{3}= 48\) feet per second.

Key concepts

Quadratic Motion Equations

Understanding quadratic motion equations is a fundamental part of calculus and physics. An object moving under the influence of gravity alone follows a quadratic relationship between the distance it travels and time. For example, in the textbook problem, we are given the equation \(y = 16t^2\), which describes the distance, \(y\), that an object falls in \(t\) seconds from rest. In this quadratic equation, the coefficient 16 represents half of the acceleration due to gravity, measured in feet per second squared, assuming no air resistance.

The general form of a quadratic motion equation can be written as \(y = at^2 + bt + c\), where \(a\) is related to acceleration, \(b\) to initial velocity, and \(c\) to the initial position. Given that objects dropped from rest have an initial velocity of zero, that part of the equation is omitted in our problem, simplifying our formula. It's crucial to distinguish this formula from linear motion equations, which have a direct proportionality between distance and time and are used for objects moving at a constant speed.

Distance-Time Relationship

The distance-time relationship in the context of our problem represents how the distance an object has fallen changes over time when dropped from rest. Specifically, the relationship is quadratic, as the distance does not increase linearly but rather is proportional to the square of the time elapsed.

By investigating this relationship, we can see important properties of the motion, such as how the speed of the object increases as time passes. Initially, at time \(t = 0\), the object is at rest; thus, it has fallen zero feet. As time progresses, the distance it falls, given by \(y = 16t^2\), grows more rapidly. This is because the object is accelerating due to gravity, leading to a steeper slope in the distance-time graph as time increases. This tells us that the speed is not constant, which is why looking at average speed over an interval is so insightful for understanding the overall behavior of the object's movement.

Problem Solving in Calculus

Problem solving in calculus often involves recognizing which mathematical concepts apply to a given real-world scenario and then utilizing the appropriate formulas. In the given problem, the first step was to identify the correct kinematic equation that models the physical situation—here, an object in free fall whose displacement adheres to a quadratic relationship with time.

Having identified the formula \(y = 16t^2\), the next step involved substituting the given time interval into the equation to find the distance fallen. This represents a direct application of the function to obtain concrete values from abstract concepts. The final step required calculating the average speed, which involves understanding that speed is a measure of distance over time. It encompasses the ability to simplify and manipulate the equation to derive the desired quantity. Effective problem solving in calculus is not only about manipulation of the numbers, but also interpreting the results in the context of the real world, such as understanding what the average speed represents for the moving object.