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The text in the image describes a mathematical problem related to velocity and acceleration. Here's the translation and the problem statement:

**Problem:**
A particle moves along a straight path. The displacement \( s \) (in meters) of the particle from a fixed point O is given by the equation \( s = t^3 - 6t^2 + 9t \), where \( t \) is the time in seconds. Find the velocity and acceleration of the particle at any time \( t \).

### Solution

1. **Velocity (\( v \)):**
   The velocity \( v \) of the particle is the first derivative of the displacement \( s \) with respect to time \( t \). Thus,
   \[
   v = \frac{ds}{dt} = \frac{d}{dt}(t^3 - 6t^2 + 9t)
   \]

2. **Acceleration (\( a \)):**
   The acceleration \( a \) of the particle is the derivative of velocity \( v \), which is the second derivative of displacement \( s \). Hence,
   \[
   a = \frac{dv}{dt} = \frac{d^2s}{dt^2}
   \]

Let's calculate these derivatives to find the expressions for velocity and acceleration.The expressions for velocity and acceleration of the particle at any time \( t \) are:

1. **Velocity (\( v \))**:
   \[
   v = 3t^2 - 12t + 9
   \]
   This represents the rate of change of displacement with respect to time.

2. **Acceleration (\( a \))**:
   \[
   a = 6t - 12
   \]
   This represents the rate of change of velocity with respect to time.

Would you like more details or have any questions on how these were derived?

### Related Questions:
1. How would you find the velocity at \( t = 2 \) seconds?
2. What is the acceleration at \( t = 3 \) seconds?
3. How would you determine the time when the particle is momentarily at rest?
4. What does it mean physically when the acceleration is zero?
5. Can you calculate the total displacement from \( t = 0 \) to \( t = 4 \) seconds?

**Tip:** To understand the motion of the particle better, consider plotting the displacement, velocity, and acceleration functions over a range of time values.

A uniform rope with a mass m and length l rotates with an angular velocity ω on a frictionless horizontal surface. One end of the rope is fixed at point O, while a small ball with mass M is attached to the other free end. Find the tension at various points in the rope.

Learn how to calculate the tension at various points in a rotating rope with a mass m and length l, with one end fixed and a ball with mass M attached to the other end. This physics problem involves concepts of circular motion, tension, and centripetal force, suitable for students in Grades 10-12.

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