Problem 44 - Calculus: Early Transcendentals

Question

A particle moves along a straight line with equation of motion \( s = f(t) \), where \( s \) is measured in meters and \( t \) in seconds. Find the velocity and the speed when \( t = 4 \). \( f(t) = 10 + \frac{45}{t + 1} \)

Step-by-Step Solution

Answer

Velocity at \( t=4 \) is \(-1.8\) m/s; Speed is \(1.8\) m/s.

Step 1: Understand the Motion Equation

The given equation of motion is \( s = f(t) = 10 + \frac{45}{t + 1} \). Here, \( s \) represents the position of the particle in meters, and \( t \) is the time in seconds. We need to find both the velocity and the speed at \( t = 4 \).

Step 2: Find the Velocity Function

The velocity \( v(t) \) is the derivative of the position function \( s = f(t) \) with respect to time \( t \). So, we differentiate \( s = 10 + \frac{45}{t + 1} \). Using the power rule and the constant rule for differentiation, we get:\[ v(t) = \frac{d}{dt} \left( 10 + \frac{45}{t + 1} \right) = 0 - \frac{45}{(t + 1)^2} = -\frac{45}{(t+1)^2} \]

Step 3: Calculate the Velocity When \( t = 4 \)

Substitute \( t = 4 \) into the velocity function to find the velocity at that moment:\[ v(4) = -\frac{45}{(4 + 1)^2} = -\frac{45}{25} = -1.8 \]The velocity of the particle at \( t = 4 \) seconds is \( -1.8 \) meters per second.

Step 4: Determine the Speed When \( t = 4 \)

Speed is the absolute value of velocity. Thus, the speed of the particle at \( t = 4 \) is:\[ \text{Speed} = |v(4)| = |-1.8| = 1.8 \]The speed is \( 1.8 \) meters per second.

Key Concepts

Understanding Velocity

Velocity is a vector quantity that describes how fast an object is moving and in which direction. It has both magnitude and direction. For our exercise, to find the velocity of the particle at any given time, we need to take the derivative of the position function, which is a technique used in calculus. This derivative gives us a new function called the velocity function, denoted as \( v(t) \). In our scenario, after differentiating the function \( s = 10 + \frac{45}{t + 1} \), we acquired the velocity function \( v(t) = -\frac{45}{(t+1)^2} \).

Velocity tells us if the particle moves forward or backward. A negative velocity means the particle is moving in the reverse direction along the line. For \( t = 4 \), the velocity is \( -1.8 \) meters per second, indicating backward motion. So, whenever you need to find velocity, remember it's about change over time, derived mathematically from position.

Defining Speed

Speed is a scalar quantity, focusing solely on how fast an object is moving. Unlike velocity, speed does not consider direction. Therefore, when you have velocity, finding speed becomes straightforward—it’s simply the absolute value of velocity.

In this exercise, the particle's velocity at \( t = 4 \) was \( -1.8 \) meters per second. To determine speed, we take the absolute value: \( |v(4)| = |-1.8| = 1.8 \) meters per second. Speed also indicates motion efficiency, essentially how much ground the particle covers, regardless of direction. This singular focus on magnitude makes speed useful in a wide range of applications, from driving to racing.

The Role of Derivative

In calculus, a derivative measures how a function changes as its input changes. Derivatives are essential for determining rates of change, such as velocity from position functions. The differentiation process allows us to create another function that expresses how one variable, like position, changes concerning another, like time.

For our exercise, the derivative of the position function \( s(t) = 10 + \frac{45}{t + 1} \) produces the velocity function \( v(t) = -\frac{45}{(t+1)^2} \). By finding this derivative, we understand not only how fast the particle is moving but also its directional trend. Calculus provides the tools to handle these changes, offering a deeper insight into the behavior of moving objects.

• Position differentiates into velocity, indicating speed and direction change.
• Always pairs calculus principles with real-world motion dynamics.

Embracing derivatives opens up a robust analytical approach, crucial for fields like physics and engineering, where understanding the subtleties of motion is vital.