1.a ball with a mass 2kg rests on an incline with an angle of 10 degrees. The ball is released to roll down the incline plane neglecting all friction:

Determine the following:

A.the loss in potential energy after it has rolled 12m.
B.the velocity after it has rolled 12m.
C.the original height that the ball has rolled from in order to reach the bottom of the slope at 20m/s

To determine the answers to these questions, we need to use principles of physics related to potential energy, kinetic energy, and the conservation of energy.

A. The loss in potential energy after it has rolled 12m:
The potential energy, PE, of an object at a height h is given by the formula PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2 on Earth), and h is the height.

In this case, the ball has a mass of 2kg, so its potential energy at the start is PE = 2 kg * 9.8 m/s^2 * h, where h is the initial height.

Since the ball rolls down the incline for 12m, we need to find the difference in height between the starting point and the end point. To calculate that difference, we can use trigonometry.

The vertical height difference between the starting point and the end point is given by h = 12m * sin(10 degrees). Plug this value into the formula for potential energy to find the loss in potential energy.

B. The velocity after it has rolled 12m:
The conservation of energy states that the total mechanical energy, E, of an object is constant, meaning the sum of its potential and kinetic energy remains the same.

At the initial height, the ball has potential energy. As it rolls down and loses height, that potential energy is converted to kinetic energy.

The potential energy, PE, at the beginning is given by PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s^2), and h is the initial height.

At the end of the 12m distance, the ball will have converted all of its potential energy into kinetic energy, KE.

The kinetic energy, KE, is given by KE = 0.5 * m * v^2, where m is the mass of the object and v is its velocity.

We can equate the initial potential energy to the final kinetic energy and solve for the velocity, v.

C. The original height that the ball has rolled from in order to reach the bottom of the slope at 20 m/s:
To determine the initial height, we can use the principle of conservation of mechanical energy. The initial potential energy at height h is equal to the final kinetic energy at the bottom of the slope.

Set the initial potential energy, PE = m * g * h, equal to the kinetic energy at the bottom, KE = 0.5 * m * v^2, where m is the mass of the object, g is the acceleration due to gravity, h is the initial height, and v is the final velocity (20 m/s). Solve for h to find the original height.

Remember to use the appropriate units for calculations.