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.