a particle of mass *m* = 2 *kg* is moving along the x-axis with a velocity *v(t)* = 3*t* - 4 m/s, where *t* is time in seconds. at *t* = 2 s, the particle is subject to a force *F(t)* = -k • *v(t)*, where *k* = 5 N • s/m is a positive constant. 

calculate the acceleration of the particle at *t = 2s*.

We are given the following information:

- Mass of the particle: \( m = 2 \, \text{kg} \)
- Velocity function: \( v(t) = 3t - 4 \, \text{m/s} \)
- Force function: \( F(t) = -k \cdot v(t) \) where \( k = 5 \, \text{N} \cdot \text{s/m} \)
- Time of interest: \( t = 2 \, \text{s} \)

### Step 1: Calculate the velocity at \( t = 2 \, \text{s} \)
To find the velocity of the particle at \( t = 2 \, \text{s} \), we substitute \( t = 2 \) into the velocity equation:

\[
v(2) = 3(2) - 4 = 6 - 4 = 2 \, \text{m/s}
\]

### Step 2: Calculate the force at \( t = 2 \, \text{s} \)
Next, use the force equation \( F(t) = -k \cdot v(t) \) to find the force at \( t = 2 \):

\[
F(2) = -5 \cdot v(2) = -5 \cdot 2 = -10 \, \text{N}
\]

### Step 3: Use Newton's Second Law to find the acceleration
Newton's second law relates force and acceleration as:

\[
F = m \cdot a
\]

Solving for acceleration:

\[
a = \frac{F}{m} = \frac{-10}{2} = -5 \, \text{m/s}^2
\]

### Final Answer:
The acceleration of the particle at \( t = 2 \, \text{s} \) is \( \boxed{-5 \, \text{m/s}^2} \).

Do you want more details, or do you have any other questions?

Here are 5 related questions:
1. How would the acceleration change if the mass were doubled?
2. What is the velocity of the particle at \( t = 3 \, \text{s} \)?
3. How would the force change if \( k \) were reduced to 3 N·s/m?
4. Can the particle come to rest? If so, when?
5. How does the velocity vary over time? Can you graph it?

**Tip:** The force \( F(t) = -k \cdot v(t) \) is a damping force, which reduces the velocity of the particle over time.

A particle of mass *m* = 2 *kg* is moving along the x-axis with a velocity *v(t)* = 3*t* - 4 m/s, where *t* is time in seconds. At *t* = 2 s, the particle is subject to a force *F(t)* = -k • *v(t)*, where *k* = 5 N • s/m is a positive constant. Calculate the acceleration of the particle at *t = 2s*.

Learn how to calculate the acceleration of a 2 kg particle moving along the x-axis with a velocity function v(t) = 3t - 4 m/s, subjected to a damping force F(t) = -k • v(t) at t = 2 seconds. This problem involves key concepts of Newton's Second Law and kinematics, suitable for advanced high school students.

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