Problem 31 - 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? \(s=200 t-16 t^{2}, \quad 0 \leq t \leq 12.5\) (a heavy object fired straight up from Earth's surface at 200 \(\mathrm{ft} / \mathrm{sec} )\)

Step-by-Step Solution

Answer

The position is maximized and changes direction at \( t = 6.25 \). Speed is 0 at this point, maximum at \( t=0 \). The body moves to the right for \( 0 \leq t < 6.25 \), left for \( t > 6.25 \). Acceleration is constant, slowing the body as \( t \) increases.

Step 1: Find the Velocity Function

The velocity function is the first derivative of the position function. Start with the position function \( s(t) = 200t - 16t^2 \). Differentiate this function with respect to \( t \) to obtain the velocity function:\[ v(t) = \frac{d}{dt}(200t - 16t^2) = 200 - 32t \]

Step 2: Determine When Velocity is Zero

Set the velocity function equal to zero to find when the body is momentarily at rest:\[ 200 - 32t = 0 \]Solving for \( t \) gives:\[ t = \frac{200}{32} = 6.25 \]

Step 3: Find the Acceleration Function

The acceleration function is the derivative of the velocity function. Differentiate the velocity function \( v(t) = 200 - 32t \):\[ a(t) = \frac{d}{dt}(200 - 32t) = -32 \]

Step 4: Graph the Functions

Graph the position function \( s(t) = 200t - 16t^2 \), the velocity function \( v(t) = 200 - 32t \), and the constant acceleration function \( a(t) = -32 \) over the given interval \( 0 \leq t \leq 12.5 \). Notice where the velocity and acceleration intersect or reach certain values.

Step 5: Analyze the Motion

- **Momentarily at Rest:** The body is at rest at \( t = 6.25 \) when \( v(t) = 0 \).- **Direction of Movement:** To the right (up) when \( v(t) > 0 \), i.e., \( 0 \leq t < 6.25 \). To the left (down) when \( v(t) < 0 \), i.e., \( t > 6.25 \).- **Change of Direction:** Occurs at \( t = 6.25 \).- **Speeding Up and Slowing Down:** The body slows down when velocity and acceleration are opposite signs, which does not occur here since acceleration is constant (negative). It speeds up when they have the same sign after \( t > 6.25 \).- **Fastest Movement:** At or just before the initial point \( t = 0 \) and just before changing direction.- **Farthest from Origin:** At the point of return at highest \( s(t) \), found by evaluating \( s(6.25) \).

Key Concepts

Position Function

The position function, often denoted as \( s(t) \), is crucial in describing how the position of a body changes over time. For our exercise, the position function is given by \( s(t) = 200t - 16t^2 \). This mathematical expression represents a heavy object being launched straight up, outlining its movement along the \( s \)-axis over time \( t \).

• The term \( 200t \) accounts for the initial upward velocity, indicating a starting speed of 200 ft/sec.
• The term \(-16t^2\) represents the gravitational pull acting on the object, slowing it down over time.

To understand the object's trajectory, we analyze the position function by graphing it. This visual representation helps us determine how far the object moves along the axis during different time intervals. Graphing shows how the function is a parabola, opening downwards due to the negative coefficient of \( t^2 \). At the vertex, the object reaches its highest point, momentarily stopping before descending back down.

Velocity and Acceleration

Velocity and acceleration are two integral metrics in motion analysis that tell us about the speed and change in speed of an object over time.

• **Velocity**: Defined as the rate of change of the position function, it is found by differentiating \( s(t) \). Thus, the velocity function for the object is \( v(t) = 200 - 32t \). This linear equation tells us how fast the object is moving and in which direction.
  - When \( v(t) > 0 \), the object moves upward (to the right). - When \( v(t) < 0 \), the object moves downward (to the left). - When \( v(t) = 0 \), the object is momentarily at rest, as seen at \( t = 6.25 \).
• **Acceleration**: This is the rate at which velocity changes over time. It's computed by differentiating the velocity function. Here, \( a(t) = -32 \), indicating a constant negative acceleration due to gravity acting on the object.

Although acceleration is constant and negative, it signifies that the object is decelerating until it stops momentarily and accelerates downward post \( t = 6.25 \). Key insights into motion changes can be analyzed by assessing where velocity intersects certain milestones.

Motion Analysis

Understanding an object's motion requires a thorough examination of its position, velocity, and acceleration over time. Motion analysis aims to comprehend when and where the object changes direction, speeds up, or slows down. In our case, the analysis reveals several insights:

• **Moment of Rest**: Occurs at \( t = 6.25 \) where \( v(t) = 0 \), marking a brief pause in motion.
• **Direction Changes**: The object shifts its trajectory at \( t = 6.25 \) transitioning from moving upwards to downwards.
• **Speed Dynamics**: Before \( t = 6.25 \), the object slows due to gravity, and post \( 6.25 \), it accelerates downward.
• **Maximum Distance from Origin**: Achieved at \( t = 6.25 \) as it's the highest vertical point in its path.

This analysis reflects the body's behavior through various positional and speed alterations. Observing these characteristics helps predict how objects will behave under similar conditions, which is pivotal in fields like physics and engineering.