Problem 34 - Thomas Calculus

Question

Exercises \(31-34\) give the position function \(s=f(t)\) of a body moving along the \(s\) -axis as a function of time \(t .\) Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t)\) . Comment on the body's behavior in relation to the signs and values of \(v\) and \(a\) . Include in your commentary such topics as the following: a. When is the body momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$ s=4-7 t+6 t^{2}-t^{3}, \quad 0 \leq t \leq 4 $$

Step-by-Step Solution

Answer

The body is at rest when \(v(t) = 0\). It moves right when \(v(t) > 0\) and left when \(v(t) < 0\). Direction changes where \(v(t)\) changes sign.

Step 1: Find the Velocity Function

The velocity function is found by taking the first derivative of the position function with respect to time t. Given \(s(t) = 4 - 7t + 6t^2 - t^3\), the velocity function is:\[v(t) = \frac{ds}{dt} = f'(t) = -7 + 12t - 3t^2\]

Step 2: Find the Acceleration Function

The acceleration function is the derivative of the velocity function with respect to time. Thus, we take the derivative of \(v(t)\) found in Step 1:\[a(t) = \frac{d^2s}{dt^2} = v'(t) = 12 - 6t\]

Step 3: Analyze the Body's Movement Part I

To determine when the body is momentarily at rest, we set the velocity function equal to zero and solve for \(t\):\[-7 + 12t - 3t^2 = 0\]Solving this quadratic equation using the quadratic formula, we get the points where the body is at rest.

Step 4: Analyze the Body's Movement Part II

To determine when the body moves left or right, analyze the sign of \(v(t)\). It moves right (up) where \(v(t) > 0\) and left (down) where \(v(t) < 0\). To find changes in the direction, check where \(v(t)\) changes sign.

Step 5: Determine Speed Changes

The body speeds up when the velocity and acceleration have the same sign (both positive or both negative) and slows down when they have opposite signs. Use \(v(t)\) and \(a(t)\) to find these intervals.

Step 6: Determine Extremes of Speed

The highest or lowest speeds occur at the endpoints of the velocity function and when the acceleration equals zero. Substitute these \(t\) values into \(|v(t)|\) to find maximum and minimum speeds.

Step 7: Find the Farthest Distance from Origin

The body is farthest from the origin when the absolute value of the position function \(|s(t)|\) is maximized. Evaluate \(s(t)\) at critical points found from \(v(t) = 0\) and endpoints of the interval to determine this number.

Key Concepts

Position Function

In calculus, the position function is a formula that describes how the position of an object changes over time. For our given problem, this function is represented by \( s(t) = 4 - 7t + 6t^2 - t^3 \). Here, \( s \) is the position of the body at any time \( t \). The coefficients \(4, -7, 6,\) and \(-1\) provide the starting point and affect how the body moves along the \(s\)-axis as time progresses.

• This polynomial equation is of the third degree, indicating it's a cubic function. This means it may have turning points and can change directions.
• The domain of the function is limited by the context of the problem, where \(0 \leq t \leq 4\), representing a specific time period over which we want to analyze the body's movement.
• Calculating the position at different values of \(t\) helps us determine the body's path, including where it starts, turns, and ends.

Velocity Function

The velocity function provides insight into the object's speed and direction at any given time. It is the derivative of the position function with respect to time \(t\). For our exercise, this function is \( v(t) = -7 + 12t - 3t^2 \).

• The velocity function tells us how fast the body is moving and whether it is moving forwards or backwards along the \(s\)-axis.
• When \(v(t) = 0\), the body is momentarily at rest. Solving this gives us the times \(t\) when the velocity is zero.
• A positive value of \(v(t)\) indicates movement to the right (or upward), while a negative value shows movement to the left (or downward).

By analyzing the sign of \(v(t)\) over different intervals, we learn when the body changes direction. The points where the sign changes indicate these transitions.

Acceleration Function

Acceleration function reflects the rate of change of the velocity function over time. It's the second derivative of the position function, or the first derivative of the velocity function, given by \( a(t) = 12 - 6t \) for our problem.

• The acceleration magnitude tells us how quickly the velocity is changing. A larger acceleration magnitude means the velocity is changing rapidly.
• When \( a(t) = 0 \), it doesn't necessarily mean the body's speed is zero; rather, it's at a point of changing acceleration.
• If \( a(t) \) and \( v(t) \) share the same sign, the body speeds up. If they have opposite signs, the body is slowing down.

Calculating this function allows us to understand the body's dynamic behavior and helps predict future position and velocity. This is vital in predicting how external forces influence movement.

Body Movement Analysis

Understanding body movement involves examining the interplay between position, velocity, and acceleration. From our step by step approach, we can discuss several critical aspects of movement.

• The body is momentarily at rest at points where \( v(t) = 0 \), identified by solving the velocity equation.
• Changes in the body's direction occur at these rest points or when \( v(t) \) changes sign.
• To figure out when it's moving fastest or slowest, we examine the absolute values of \(v(t)\), particularly at endpoints and where \(a(t) = 0\).
• The body moves farthest from its starting point where the position function's value is at its maximum or minimum within the considered domain.
• The interplay between \(v(t)\) and \(a(t)\) further reveals when the body speeds up or slows down.

Examining these data points and understanding their interactions in terms of calculus provides a comprehensive picture of the body's movement characteristics.