Problem 13 - Calculus Single Variable

Question

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=6 t+3 $$

Step-by-Step Solution

Answer

At time \( t=4 \), the body is moving forward.

Step 1: Understand the Position Function

The position of the moving body is given by the function \( p(t) = 6t + 3 \). This is a linear equation representing the body's position at any time \( t \). The slope of this line (the coefficient of \( t \)) will help us determine the direction of movement.

Step 2: Determine Velocity from Position

To find out in which direction the body is moving, we need to calculate its velocity, which is the derivative of the position function with respect to time. The derivative of \( p(t) = 6t + 3 \) with respect to \( t \) is \( v(t) = \frac{d}{dt}(6t + 3) = 6 \).

Step 3: Analyze the Velocity at t = 4

The velocity function \( v(t) = 6 \) is constant and does not depend on \( t \). Therefore, at \( t = 4 \), the velocity \( v(4) = 6 \). Since the velocity is positive, the body is moving forward.

Key Concepts

Position Function

The position function, denoted as \( p(t) \), describes the location of a moving body at any given time \( t \). When we have a position function like \( p(t) = 6t + 3 \), it tells us how the position of the body changes over time. Here, \( t \) represents time, and the function gives us the position as a linear equation.

• In this equation, 6 represents the rate of change, indicating how fast the position is changing with time.
• The number 3 is the starting position of the body at \( t = 0 \).

This simple linear function is easy to read: as time advances, the position increases straight along a path determined by the slope, indicating the body's consistent pace.

Derivative

The derivative is a mathematical concept that helps us find the rate at which a function is changing. When considering the position function \( p(t) = 6t + 3 \), taking the derivative with respect to time \( t \) gives us the velocity function.

• To calculate the derivative, we differentiate each term in the position function. The derivative of \( 6t \) is 6, and the derivative of any constant like 3 is 0.
• Thus, the derivative of the position function is \( v(t) = \frac{d}{dt}(6t + 3) = 6 \).

This derivative reveals the relationship between time and position. Here, the derivative \( v(t) = 6 \) is a constant, which implies that the position changes uniformly over time.

Linear Motion

Linear motion involves movement along a straight path with a constant speed. This is reflected in our position function \( p(t) = 6t + 3 \). The body's motion described by this function is linear because:

• The position changes at a constant rate, indicated by the slope 6.
• The path of the motion is straight, which is characteristic of linear equations.

Therefore, in this linear motion scenario, the body continues to move forward consistently in a straight line, which is also evident from the constant velocity derived from the function.

Direction Analysis

Direction analysis involves determining which way a body is moving based on its velocity. In our scenario, the body described by the position function \( p(t) = 6t + 3 \) has a velocity function \( v(t) = 6 \). Here are the insights from analyzing this velocity:

• A positive velocity, like 6, indicates that the body is moving forward.
• Since \( v(t) \) is constant and positive for all \( t \), the body never changes its direction and keeps moving in the same direction.
• At \( t = 4 \), the velocity is still 6, confirming the body is moving forward.

Direction analysis uses the velocity to understand motion behavior, and in this case, the consistent positive velocity means ongoing forward motion.