Question

A ball rolls down a long inclined plane so that its distance \(s\) from its starting point after \(t\) seconds is \(s=4.5 t^{2}+2 t\) feet. When will its instantaneous velocity be 30 feet per second?

Short answer

The instantaneous velocity is 30 feet per second when \( t \approx 3.11 \) seconds.

Step-by-step solution

Understand the Problem

We have a function \( s(t) = 4.5t^2 + 2t \) which gives us the distance \( s \) that a ball rolls down an inclined plane as a function of time \( t \). We need to find when its instantaneous velocity is 30 feet per second.

Find the Velocity Function

The velocity of the ball is the derivative of the distance function \( s(t) \) with respect to time \( t \). So, we need to differentiate \( s(t) = 4.5t^2 + 2t \).The derivative is:\[ v(t) = \frac{ds}{dt} = \frac{d}{dt}(4.5t^2 + 2t) = 9t + 2 \]Thus, the velocity function is \( v(t) = 9t + 2 \).

Set up the Equation for Instantaneous Velocity

The problem asks when the instantaneous velocity is 30 feet per second. We set the velocity function equal to 30:\[ 9t + 2 = 30 \]

Solve for Time \( t \)

Solve the equation \( 9t + 2 = 30 \) for \( t \).Subtract 2 from both sides:\[ 9t = 28 \]Now divide by 9:\[ t = \frac{28}{9} \approx 3.11 \]Therefore, \( t \approx 3.11 \) seconds.

Key concepts

Instantaneous Velocity

Instantaneous velocity refers to the speed and direction of an object at a specific moment in time. Unlike average velocity, which is calculated over a period of time, instantaneous velocity is like taking a snapshot of movement. For a rolling ball described by a function of time, such as the one in our exercise, you can find its instantaneous velocity by looking at its velocity at a precise instance.
This concept is crucial when you need to know how fast something is moving exactly at a certain point in time, for example, during a particular segment of its journey.

• **Instantaneous Velocity Formula**: It is derived from the derivative of the position function, representing the object's movement.
• **Example in Exercise**: For our ball, the position function is given as \( s(t) = 4.5t^2 + 2t \). The instantaneous velocity at any time \( t \) is found by differentiating this position function.
• **Calculation**: The velocity function \( v(t) = 9t + 2 \) can be used to find the velocity at an exact time, such as when it equals 30 feet per second.

Understanding instantaneous velocity is essential in physics and engineering, where knowing the speed at a specific second can affect the system's design or control.

Differentiation

Differentiation is the mathematical process used to find the derivative of a function. It's a fundamental tool in calculus that allows us to determine how a function's output changes with respect to one of its inputs. By performing differentiation, we are able to calculate rates of change, such as velocity and acceleration.
In our example, the function describes the motion of a ball, and differentiation helps us find its instantaneous velocity.

• **Purpose of Differentiation**: It helps to understand the trend or rate at which things are changing. This technique is frequently used to find the slope of the curve of a function at any point.
• **Example Application**: The position function for the ball is \( s(t) = 4.5t^2 + 2t \). By differentiating, we discover \( v(t) = 9t + 2 \).
• **Importance in Physics**: Differentiation is vital as it provides us tools to model real-world phenomena accurately, capturing how quantities like speed and energy change over time.

This process is a cornerstone of calculus and is applicable in numerous fields such as economics, statistics, and natural sciences.

Mathematical Modeling

Mathematical modeling involves creating an abstract representation of a real-world scenario through mathematical language and symbols. The objective is to simulate systems to predict and analyze their behavior under various conditions. In our exercise, the distance function of the ball is a simple yet effective model that predicts its behavior on an inclined plane over time.
By utilizing functions and their derivatives, mathematical modeling provides a way to forecast phenomena accurately.

• **Role in Problem Solving**: Mathematical models help to simplify complex systems into understandable expressions, allowing us to make informed predictions.
• **Real-Life Application**: Our exercise uses the function \( s(t) = 4.5t^2 + 2t \) to predict distances, and differentiates it to predict velocities.
• **Broader Use Cases**: Beyond physics, mathematical modeling is used in finance for modeling market trends, in biology for population dynamics, and in engineering for structural analysis.

The ability to distill a scenario into a mathematical model is invaluable for scientists and engineers when analyzing data or strategizing future developments.