A ball with a mass of 2kg rests on an angle of 10degrees .The ball is to be 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 kinetic energy after it has rolled 12m, c The velocity after it has rolled 12m, d The original height that the ball has rolled from, inorder to reach the bottom of the slope at 20m/s

To solve this problem, we can break it down into several steps.

Step 1: Find the initial potential energy of the ball.
The potential energy (PE) of an object is given by the formula PE = mgh, where m is the mass (2kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the ball above a reference point. In this case, since the ball is resting on an angle of 10 degrees, h can be calculated using trigonometry.

h = sin(10°) × d, where d is the original height that the ball has rolled from.

Step 2: Find the loss in potential energy after the ball has rolled 12m.
To find the loss in potential energy, we need to find the final height of the ball after it has rolled 12m. We can use trigonometry again to find this height.

h_final = h - 12m × sin(10°)

The loss in potential energy (ΔPE) is then given by ΔPE = mgh - mgh_final.

Step 3: Find the kinetic energy after the ball has rolled 12m.
The kinetic energy (KE) of an object is given by the formula KE = 0.5 × mv², where m is the mass of the ball (2kg) and v is its velocity.

Step 4: Find the velocity of the ball after it has rolled 12m.
To find the velocity of the ball, we can use the formula v = √(2gh), where g is the acceleration due to gravity and h is the final height of the ball.

v = √(2gh_final)

Let's calculate each part step-by-step:

Step 1: Calculate the original height (h):
h = sin(10°) × d

Step 2: Calculate the loss in potential energy (ΔPE):
h_final = h - 12m × sin(10°)
ΔPE = mgh - mgh_final

Step 3: Calculate the kinetic energy (KE):
KE = 0.5 × mv²

Step 4: Calculate the velocity (v):
v = √(2gh_final)

Well, isn't this a rolling good question? Let's roll with it and find some answers!

a) To determine the loss in potential energy after rolling 12m, we'll need to calculate the change in height. Considering the angle of 10 degrees and the horizontal distance rolled, we can use trigonometry to find the vertical distance, which is 12m multiplied by the sine of 10 degrees. The loss in potential energy is then given by mass multiplied by the gravitational acceleration multiplied by the change in height.

b) The kinetic energy after rolling 12m will be the total energy at that point, which is equal to the initial potential energy minus the loss in potential energy. Since we have already calculated the loss in potential energy, we can subtract it from the initial potential energy to get the kinetic energy.

c) To calculate the velocity of the ball after rolling 12m, we can use the principle of conservation of energy. The total energy of the ball is equal to the sum of its kinetic energy and potential energy. Since we have already calculated the kinetic energy after rolling 12m, we can equate the total energy to the kinetic energy and solve for velocity.

d) Now, in order for the ball to reach the bottom of the slope at 20m/s, we need to figure out the initial height it was rolled from. We can use a combination of kinematic equations and energy conservation to find this height.

Now, let's not get too tilted with these calculations and crunch some numbers!

To determine the answers to these questions, we need to use some basic principles of physics, namely gravitational potential energy, kinetic energy, and the relationship between energy and work.

a) The loss in potential energy can be calculated using the formula:

Loss in potential energy = m * g * h

where:
m = mass of the ball (2kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height or vertical distance (12m in this case)

Substituting the given values:

Loss in potential energy = 2kg * 9.8 m/s^2 * 12m

b) The kinetic energy after it has rolled 12m can be calculated using the conservation of mechanical energy:

Kinetic energy = Loss in potential energy

Since there is no friction, all the potential energy is converted to kinetic energy as the ball rolls down the incline.

c) The velocity after it has rolled 12m can be calculated using the principle of conservation of energy:

Kinetic energy = (1/2) * m * v^2

where:
m = mass of the ball (2kg)
v = velocity of the ball

Rearranging the equation, we can solve for v:

v = sqrt((2 * Kinetic energy) / m)

Substituting the calculated value of Kinetic energy from part b into the equation above will give the velocity.

d)To find the original height from which the ball has rolled to reach the bottom of the slope at 20m/s, we need to consider the conservation of mechanical energy. Initially, the ball has only potential energy, and at the bottom of the slope, all the potential energy is converted to kinetic energy.

Potential energy at the starting height = Kinetic energy at the bottom

Using the formula for potential energy:

m * g * h_start = (1/2) * m * v^2

Canceling out the mass:

g * h_start = (1/2) * v^2

Rearranging the equation to solve for h_start:

h_start = (1/2) * v^2 / g

Substituting the given velocity of 20 m/s and acceleration due to gravity of 9.8 m/s^2 will give you the answer.

Remember to double-check all calculations and units to ensure accuracy.