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 speed (in m/s) when t = 2.
f(t) = 12 + 15/t + 1
velocity	
 m/s
speed	
 m/s

To find the velocity and speed of a particle at a specific time \( t = 2 \), we need to first differentiate the position function \( s = f(t) \) to obtain the velocity function, and then evaluate it at \( t = 2 \).

Given the position function:

\[
f(t) = 12 + \frac{15}{t + 1}
\]

### Step 1: Find the Velocity Function

The velocity \( v(t) \) is the derivative of the position function \( f(t) \) with respect to time \( t \):

\[
v(t) = \frac{d}{dt} \left( 12 + \frac{15}{t + 1} \right)
\]

Let's differentiate this function.

\[
v(t) = 0 + 15 \cdot \frac{d}{dt} \left( \frac{1}{t + 1} \right)
\]

Using the derivative rule \( \frac{d}{dt} \left( \frac{1}{t + 1} \right) = -\frac{1}{(t + 1)^2} \):

\[
v(t) = 15 \cdot \left( -\frac{1}{(t + 1)^2} \right) = -\frac{15}{(t + 1)^2}
\]

### Step 2: Evaluate the Velocity at \( t = 2 \)

Substitute \( t = 2 \) into the velocity function:

\[
v(2) = -\frac{15}{(2 + 1)^2} = -\frac{15}{3^2} = -\frac{15}{9} = -\frac{5}{3} \, \text{m/s}
\]

### Step 3: Find the Speed

The speed is the absolute value of the velocity:

\[
\text{Speed} = |v(2)| = \left| -\frac{5}{3} \right| = \frac{5}{3} \, \text{m/s}
\]

### Final Answers

- **Velocity at \( t = 2 \)**: \(-\frac{5}{3} \, \text{m/s}\)
- **Speed at \( t = 2 \)**: \(\frac{5}{3} \, \text{m/s}\)

Would you like further details or have any questions?

### Related Questions:

1. What is the acceleration of the particle at \( t = 2 \)?
2. How does the velocity change as \( t \) increases?
3. What is the position of the particle at \( t = 2 \)?
4. How would you find the maximum or minimum velocity of the particle?
5. How does the speed compare to velocity in general motion?

**Tip:** Remember, speed is always positive because it is the magnitude of velocity!

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 speed (in m/s) when t = 2. f(t) = 12 + 15/t + 1.

Learn how to find the velocity and speed of a particle moving along a straight line with the position function f(t) = 12 + 15/(t + 1). This problem involves differentiating the position function to obtain the velocity, then calculating the speed as the magnitude of velocity. Suitable for Grades 11-12 students learning calculus.

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