The potential energy (in SI units) of a particle of mass `2kg` in a conservative field is `U=6x-8y`. If the initial velocity of the particle is `vecu=-1.5hati+2hatj`, then find the total distance travelled by the particle in the first two seconds.

Explanation

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step 1: Determine the Force from Potential Energy
The potential energy \( U \) is given as:
\[ U = 6x - 8y \]
To find the force \( \vec{F} \), we use the relationship between force and potential energy:
\[ \vec{F} = -\nabla U = -\left(\frac{\partial U}{\partial x} \hat{i} + \frac{\partial U}{\partial y} \hat{j}\right) \]

Calculating the partial derivatives:
\[ \frac{\partial U}{\partial x} = 6 \quad \text{and} \quad \frac{\partial U}{\partial y} = -8 \]
Thus, the force vector becomes:
\[ \vec{F} = -\left(6 \hat{i} - 8 \hat{j}\right) = -6 \hat{i} + 8 \hat{j} \]

### Step 2: Calculate the Acceleration
Using Newton's second law, we can find the acceleration \( \vec{a} \):
\[ \vec{a} = \frac{\vec{F}}{m} \]
Given that the mass \( m = 2 \, \text{kg} \):
\[ \vec{a} = \frac{-6 \hat{i} + 8 \hat{j}}{2} = -3 \hat{i} + 4 \hat{j} \]

### Step 3: Initial Velocity
The initial velocity \( \vec{u} \) is given as:
\[ \vec{u} = -1.5 \hat{i} + 2 \hat{j} \]

### Step 4: Determine the Distance Traveled
Since the acceleration is constant and the motion is in a straight line, we can use the equation of motion:
\[ \vec{s} = \vec{u} t + \frac{1}{2} \vec{a} t^2 \]
We need to calculate the distance traveled in the first 2 seconds (\( t = 2 \, \text{s} \)):
\[ \vec{s} = \left(-1.5 \hat{i} + 2 \hat{j}\right)(2) + \frac{1}{2} \left(-3 \hat{i} + 4 \hat{j}\right)(2^2) \]

Calculating each term:
1. Initial velocity term:
\[ \vec{u} t = (-1.5 \hat{i} + 2 \hat{j})(2) = -3 \hat{i} + 4 \hat{j} \]

2. Acceleration term:
\[ \frac{1}{2} \vec{a} t^2 = \frac{1}{2} (-3 \hat{i} + 4 \hat{j})(4) = (-6 \hat{i} + 8 \hat{j}) \]

Combining both terms:
\[ \vec{s} = (-3 \hat{i} + 4 \hat{j}) + (-6 \hat{i} + 8 \hat{j}) = (-9 \hat{i} + 12 \hat{j}) \]

### Step 5: Calculate the Magnitude of the Displacement
To find the total distance traveled, we calculate the magnitude of the displacement vector:
\[ |\vec{s}| = \sqrt{(-9)^2 + (12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \, \text{m} \]

### Final Answer
The total distance traveled by the particle in the first two seconds is:
\[ \boxed{15 \, \text{m}} \]