https://file.aiquickdraw.com/mathbot/file/1727886326456ru4e9wy0.jpg

The problem is given in Ukrainian, and it can be translated as follows:

**Problem Statement:**

A material point with a mass of \( 3 \, \text{kg} \) moves in a straight line according to the law:
\[
S(t) = 4t^2 - 6t + 5
\]
Find the kinetic energy at time \( t = 2 \).

**Formula for kinetic energy:**

\[
E_k = \frac{mv^2}{2}
\]

### Solution:

1. **Determine the velocity**: The velocity \( v(t) \) is the derivative of the displacement \( S(t) \) with respect to time \( t \):
\[
v(t) = \frac{dS(t)}{dt} = \frac{d}{dt}(4t^2 - 6t + 5)
\]
Calculating the derivative:
\[
v(t) = 8t - 6
\]

2. **Find the velocity at \( t = 2 \)**:
\[
v(2) = 8 \times 2 - 6 = 16 - 6 = 10 \, \text{m/s}
\]

3. **Calculate the kinetic energy** using the formula \( E_k = \frac{mv^2}{2} \):
\[
E_k = \frac{3 \times 10^2}{2} = \frac{3 \times 100}{2} = 150 \, \text{J}
\]

Thus, the kinetic energy at \( t = 2 \) is:
\[
E_k = 150 \, \text{J}
\]

Would you like a detailed explanation of any step, or have any questions?

### Related Questions:
1. What is the acceleration of the material point at any given time \( t \)?
2. How does the kinetic energy change as \( t \) increases?
3. What is the total distance traveled by the point between \( t = 0 \) and \( t = 2 \)?
4. How would the kinetic energy change if the mass were doubled?
5. What is the significance of the initial displacement \( S(0) \)?

### Tip:
When analyzing motion problems, always start by finding the velocity and acceleration functions, as they often provide key insights into the dynamics of the system.

A material point with a mass of 3 kg moves in a straight line according to the law S(t) = 4t^2 - 6t + 5. Find the kinetic energy at time t = 2.

Find the kinetic energy of a 3 kg material point moving according to the law S(t) = 4t^2 - 6t + 5 at time t = 2. This problem covers key physics and calculus concepts, including kinematics and kinetic energy calculations. Suitable for Grades 9-12.

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