The displacement (in meters) of a particle moving in a straight line is given by the equation of motion

*s* =

6

*t*2

,

where *t* is measured in seconds. Find the velocity (in m/s) of the particle at times

*t* = *a*,

*t* = 1,

*t* = 2,

and

*t* = 3.

The displacement \( s(t) \) of a particle is given by the equation:

\[
s(t) = 6t^2
\]

The velocity \( v(t) \) of the particle is the derivative of the displacement function \( s(t) \) with respect to time \( t \). So, we find \( v(t) \) by differentiating \( s(t) \).

\[
v(t) = \frac{ds}{dt} = \frac{d}{dt} \left( 6t^2 \right) = 12t
\]

Now, we can find the velocity at different times.

1. At \( t = a \):
\[
v(a) = 12a
\]

2. At \( t = 1 \):
\[
v(1) = 12 \times 1 = 12 \ \text{m/s}
\]

3. At \( t = 2 \):
\[
v(2) = 12 \times 2 = 24 \ \text{m/s}
\]

4. At \( t = 3 \):
\[
v(3) = 12 \times 3 = 36 \ \text{m/s}
\]

### Summary of velocities:
- \( v(a) = 12a \)
- \( v(1) = 12 \ \text{m/s} \)
- \( v(2) = 24 \ \text{m/s} \)
- \( v(3) = 36 \ \text{m/s} \)

Would you like more details or have any questions?

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**Relative Questions:**
1. How do we interpret the derivative of a function in the context of motion?
2. What is the significance of the constant coefficient in the velocity equation?
3. How would you find the acceleration of the particle at a given time?
4. What happens to the velocity as \( t \) increases?
5. Can we use the same process to find displacement from velocity?

**Tip:** Derivatives are essential in physics, as they often represent rates of change, such as velocity from displacement or acceleration from velocity.

The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 6t^2, where t is measured in seconds. Find the velocity (in m/s) of the particle at times t = a, t = 1, t = 2, and t = 3.

Find the velocity of a particle moving along a straight line with displacement given by s = 6t^2. Learn how to apply differentiation to solve for velocity at t = a, t = 1, t = 2, and t = 3. Suitable for students in Grades 9-12 studying kinematics and differentiation.

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