The potential energy U in joule of a particle of mass 1kg moving in x-y plane obeys the law U = 3x + 4y, where (x,y) are the coordinates of the particle in metre. If the particle is at rest at (6,4) at time t=0, then:

• A. The particle has constant acceleration
• B. The particle has zero acceleration
• C. The speed of the particle when it crosses the y-axis is 10m/s
• D. Coordinates of the particle at t=1 sec are (4.5,2)

The potential energy of the particle is given by the equation U = 3x + 4y. To analyze the motion of the particle, we can derive the force acting on it from the potential energy. The force can be calculated using the negative gradient of the potential energy.

Calculating the Force

The force components in the x and y directions are:

• F_x = -∂U/∂x = -3
• F_y = -∂U/∂y = -4

This means the force acting on the particle is constant, specifically F = (-3, -4).

Determining Acceleration

Using Newton's second law, F = ma, we can find the acceleration:

• a_x = F_x/m = -3/1 = -3 m/s²
• a_y = F_y/m = -4/1 = -4 m/s²

Since the acceleration is constant, the correct answer is A. The particle has constant acceleration.

Analyzing Other Options

Now, let's briefly evaluate the other options:

• B. The particle has zero acceleration: This is incorrect because we found the acceleration is constant and non-zero.
• C. The speed of the particle when it crosses the y-axis is 10 m/s: This requires further calculations to verify.
• D. Coordinates of the particle at t=1 sec are (4.5, 2): This also needs to be checked against the equations of motion.

Conclusion

Based on our analysis, the particle exhibits constant acceleration. Further calculations are needed to confirm the speed at the y-axis and the coordinates at t=1 second.