1. Analyze the Forces
* Gravity: The force of gravity acting on the box is mg, where m is the mass (2 kg) and g is the acceleration due to gravity (9.8 m/s²). This force acts vertically downward.
* Normal Force: The plane exerts a force perpendicular to its surface, which we call the normal force (N).
* Friction: There are two possibilities:
* Static Friction: This force opposes the impending motion of the box and acts parallel to the plane. It's maximum value is μs * N (where μs is the coefficient of static friction).
* Kinetic Friction: This force acts parallel to the plane and opposes the motion of the box once it's moving. Its value is μk * N (where μk is the coefficient of kinetic friction).
2. Resolve Forces
* Resolve Gravity: We need to find the components of gravity parallel and perpendicular to the plane.
* Parallel component (mg sin 30°): This component pulls the box down the incline.
* Perpendicular component (mg cos 30°): This component presses the box against the plane.
* Normal Force: The normal force is equal in magnitude and opposite in direction to the perpendicular component of gravity: N = mg cos 30°.
3. Determine if the Box Moves
* Static Friction: Calculate the maximum static friction force: μs * N = 0.5 * (2 kg * 9.8 m/s² * cos 30°) ≈ 8.49 N.
* Force Down the Incline: Calculate the component of gravity pulling the box down the incline: (2 kg * 9.8 m/s² * sin 30°) = 9.8 N.
* Comparison: The force pulling the box down the incline (9.8 N) is greater than the maximum static friction force (8.49 N). This means the box will overcome static friction and start to move.
4. Calculate Acceleration
* Kinetic Friction: Now that the box is moving, we use the coefficient of kinetic friction. The kinetic friction force is μk * N = 0 * (2 kg * 9.8 m/s² * cos 30°) = 0 N.
* Net Force: The only force acting on the box down the incline is the component of gravity (9.8 N).
* Acceleration: Using Newton's Second Law (F = ma), we find the acceleration: a = F/m = 9.8 N / 2 kg = 4.9 m/s².
5. Calculate Final Velocity
* Initial Velocity: The box starts from rest, so the initial velocity (v₀) is 0 m/s.
* Time: The time is given as 3 seconds.
* Final Velocity: Using the equation v = v₀ + at, we get:
v = 0 m/s + (4.9 m/s²) * (3 s) = 14.7 m/s
Therefore, the speed of the box after 3 seconds is 14.7 m/s.
