Problem 42 - Calculus for Biology and Medicine

Question

4Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds and \(s(t)\) is measured in meters. (a) Graph \(s(t)\) for \(t \geq 0\). (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time \(0 ?\) (ii) Is there another time at which the particle visits the location where it was at time \(0 ?\) (iii) How far to the right on the straight line does the particle travel? (iv) How far to the left on the straight line does the particle travel? (v) Where is the velocity positive? where negative? equal to 0 ? (c) Find the velocity of the particle. (d) When is the velocity of the particle equal to \(1 \mathrm{~m} / \mathrm{s}\) ?

Step-by-Step Solution

Answer

At time 0, the particle is at 0 meters. It revisits 0 meters at 3 seconds, travels 2.25 meters right, and doesn't travel left. Velocity is positive for \(t < 1.5\), negative for \(t > 1.5\), and zero at \(t = 1.5\). Velocity equals 1 m/s at \(t = 1\) second.

Step 1: Graphing the Position Function

To graph the function \(s(t) = 3t - t^2\), we identify it as a quadratic function in standard form \(s(t) = at^2 + bt + c\), where \(a = -1\), \(b = 3\), and \(c = 0\). Plot key points by evaluating \(s(t)\) at various values of \(t \geq 0\). The function is a downward opening parabola because \(a < 0\). Sketch the graph using points such as \(s(0) = 0\), \(s(1) = 2\), and \(s(3) = 0\) to get an idea of its shape.

Step 2: Analyzing the Graph for Position at Time 0

Evaluate \(s(0) = 3(0) - 0^2 = 0\). Thus, the particle is at the origin \(0\) meters at time \(t = 0\).

Step 3: Identifying Repeat Positions

Look for another time \(t\) where \(s(t) = 0\). Solving \(3t - t^2 = 0\), factored as \(t(3 - t) = 0\), gives \(t = 0\) or \(t = 3\). Therefore, the particle is again at 0 meters at \(t = 3\) seconds.

Step 4: Determining Maximum Distance to the Right

Find the vertex of the parabola to determine the maximum distance traveled. The vertex of \(s(t) = -t^2 + 3t\) occurs at \(t = \frac{-b}{2a} = \frac{3}{2}\). Substitute into the function: \(s\left(\frac{3}{2}\right) = 3\cdot\frac{3}{2} - \left(\frac{3}{2}\right)^2 = \frac{9}{4}\) meters. This is the farthest point to the right.

Step 5: Determining Maximum Distance to the Left

From Steps 2 and 3, the particle returns to 0 meters at \(t = 3\). Therefore, the farthest left is 0 meters as it doesn't travel left from the origin based on the graph.

Step 6: Analyzing Velocity

The velocity function is the derivative of the position function. Compute the derivative: \(v(t) = \frac{ds}{dt} = 3 - 2t\). The velocity is positive when \(3 - 2t > 0\), which simplifies to \(t < 1.5\), zero at \(t = 1.5\), and negative when \(t > 1.5\).

Step 7: Finding Specific Velocity Condition

Set the velocity equation \(v(t) = 3 - 2t = 1\). Solve for \(t\): \(3 - 2t = 1\) gives \(2t = 2\), leading to \(t = 1\). Thus, the velocity is \(1 \text{ m/s}\) at \(t = 1\) second.

Key Concepts

Particle Motion

Understanding particle motion involves analyzing how an object moves along a path over time. In this exercise, we're dealing with a particle moving in a straight line, which simplifies our study. This means we only focus on its movement in one dimension, either forward or backward. In one-dimensional motion, a particle's path can often be described by a mathematical function, representing its position in relation to time. For this particular particle, the position function is given as \( s(t) = 3t - t^2 \), revealing the particle's location on a straight line over time \( t \geq 0 \). By examining the position function, we gain insights into where the particle is at any given moment, how far it moves to the right or left, and how its speed changes over time.

Velocity

Velocity is a key concept in understanding the motion of a particle. It tells us not just how fast the particle is moving, but also its direction. In this problem, to find the velocity, we need to compute the derivative of the position function \( s(t) = 3t - t^2 \). The velocity function is thus given by \( v(t) = \frac{ds}{dt} = 3 - 2t \). This equation provides a dynamic view of the particle's motion. Some key aspects to consider include:

• When \( 3 - 2t > 0 \), the particle has positive velocity, indicating it's moving to the right.
• When \( 3 - 2t < 0 \), the velocity is negative, showing it's moving to the left.
• At \( t = 1.5 \), the velocity is zero, meaning the particle momentarily stops before reversing direction.

These intervals are essential for understanding how the particle's speed and direction evolve over its path.

Quadratic Functions

Quadratic functions like \( s(t) = 3t - t^2 \) are central to the study of motion, especially in this exercise. A quadratic function is of the form \( at^2 + bt + c \). Here, the coefficients \( a = -1 \), \( b = 3 \), and \( c = 0 \) reveal a lot about the function's characteristics. The negative \( a \) means the parabola opens downwards, helping in visualizing the path of motion. By identifying the vertex formula \( t = \frac{-b}{2a} \), we find this occurs at \( t = 1.5 \). Substituting back, the maximum point reached by the particle is \( \frac{9}{4} \) meters. Quadratics help model the ups and downs in physical scenarios like projectile motion or, in this case, particle motion.

Position Function

The position function \( s(t) = 3t - t^2 \) provides a complete description of the particle's location over time. It's a tool for solving various questions about the motion. For example:

• At \( t = 0 \), \( s(0) = 0 \), showing the particle starts at the origin.
• At \( t = 3 \), \( s(3) = 0 \) again, indicating the particle returns to the starting point.

The position function's graph is a parabola opening downwards, and it's useful for deducing maximum and minimum positions. By examining points like \( s(1) = 2 \), where the particle has moved furthest to the right, we gain a deeper understanding of the entire motion. The position function not only aids in graphing the path but also in predicting where the particle will be at certain times.