Problem 253 - Calculus Volume 1

Question

The position function of a freight train is given by \(s(t)=100(t+1)^{-2},\) with \(s\) in meters and \(t\) in seconds. At time \(t=6 \mathrm{~s}\), find the train's a. velocity and b. acceleration. c. Using a. and b. is the train speeding up or slowing down?

Step-by-Step Solution

Answer

a. Velocity is approximately -0.583 m/s. b. Acceleration is approximately 0.250 m/s². c. The train is slowing down.

Step 1: Differentiate for Velocity

The velocity function is the derivative of the position function with respect to time, given as \( s(t) = 100(t+1)^{-2} \). To find \( v(t) \), differentiate the position function: \( v(t) = \frac{d}{dt}[100(t+1)^{-2}] = -200(t+1)^{-3} \).

Step 2: Calculate Velocity at t=6

Using the velocity function \( v(t) = -200(t+1)^{-3} \), substitute \( t = 6 \) to find the velocity: \( v(6) = -200(6+1)^{-3} = -200 \times 7^{-3} = -\frac{200}{343} \approx -0.583 \text{ m/s} \).

Step 3: Differentiate for Acceleration

The acceleration function is the derivative of the velocity function with respect to time. To find \( a(t) \), differentiate \( v(t) = -200(t+1)^{-3} \): \( a(t) = \frac{d}{dt}[-200(t+1)^{-3}] = 600(t+1)^{-4} \).

Step 4: Calculate Acceleration at t=6

Using the acceleration function \( a(t) = 600(t+1)^{-4} \), substitute \( t = 6 \) to find the acceleration: \( a(6) = 600(7)^{-4} = \frac{600}{2401} \approx 0.250 \text{ m/s}^2 \).

Step 5: Determine if the Train is Speeding Up or Slowing Down

Check the signs of velocity and acceleration at \( t = 6 \). The velocity is negative (\(-0.583 \text{ m/s} \)) and the acceleration is positive (\(0.250 \text{ m/s}^2 \)). Since the signs are opposite, the train is slowing down.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which evaluates the rate at which the function's value changes with respect to a change in its input. In the context of our position function for the freight train, differentiation helps us find the velocity and acceleration functions to describe the train's motion.

• The position function given is \( s(t) = 100(t+1)^{-2} \). This represents the train's position in meters at any time \( t \) in seconds.
• To find the velocity function or \( v(t) \), we take the derivative of the position function \( s(t) \) with respect to \( t \). This leads to \( v(t) = -200(t+1)^{-3} \).
• Further differentiation of \( v(t) \) provides the acceleration function \( a(t) \), expressed as \( a(t) = 600(t+1)^{-4} \).

These derivatives are key in analyzing the train's movement, allowing us to compute specific values like velocity and acceleration at any given time.

Velocity and Acceleration

In physical terms, velocity is the measure of how fast and in what direction an object moves, while acceleration is the rate of change of velocity. For the freight train described by the position function \( s(t) = 100(t+1)^{-2} \), finding the velocity and acceleration involves specific calculations using differentiation.
Let's focus on how each is determined:

• Velocity: The velocity function \( v(t) = -200(t+1)^{-3} \) was obtained by differentiating the position function. By substituting \( t = 6 \) into this function, we get \( v(6) = -\frac{200}{343} \approx -0.583 \, \text{m/s} \), indicating that the train moves backwards, as suggested by the negative value.
• Acceleration: The function \( a(t) = 600(t+1)^{-4} \) indicates the rate of change of the velocity. Substituting \( t = 6 \) yields \( a(6) = \frac{600}{2401} \approx 0.250 \, \text{m/s}^2 \). A positive acceleration suggests an increase in velocity in the direction of motion.

Hence, these calculations provide insights into the real-time motion dynamics of the train.

Position Function Analysis

Analyzing a position function allows us to understand how an object's location changes over time. In our case, the position function \( s(t) = 100(t+1)^{-2} \) describes how the freight train's position changes as time progresses.
Let's break down why analyzing this is essential:

• Understanding Motion: This function shows the position is inversely dependent on \((t+1)^2\), meaning as time increases, the train's position approaches zero, likely representing the train moving towards a specific location.
• Derivatives Inform Motion Dynamics: By calculating first and second derivatives (velocity and acceleration), we gather detailed insights on how the train speeds up or slows down.
• Practical Application: These insights can be crucial for engineers and physicists who need to ensure the train runs efficiently and safely.

Using this analysis, we observed at \( t = 6 \) seconds, the train is slowing down, as it has negative velocity but positive acceleration.