A ball rolls down a ramp of height h. If it starts from rest, what is its final velocity at the bottom?

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Q: A ball rolls down a ramp of height h. If it starts from rest, what is its final velocity at the bottom?

Solution: Using conservation of energy, the final velocity v at the bottom is given by v = √(2gh).

Steps: 10

Step 1: Understand that the ball starts from rest, which means its initial velocity is 0.
Step 2: Recognize that the ball is rolling down a ramp, which means it is losing potential energy and gaining kinetic energy.
Step 3: Identify the height of the ramp as 'h'. This height represents the potential energy the ball has at the top.
Step 4: Use the principle of conservation of energy, which states that the total energy at the top equals the total energy at the bottom.
Step 5: At the top, the ball has potential energy given by the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height.
Step 6: At the bottom, all the potential energy has converted into kinetic energy (KE), which is given by the formula KE = 0.5 * m * v^2, where v is the final velocity.
Step 7: Set the potential energy equal to the kinetic energy: mgh = 0.5 * m * v^2.
Step 8: Cancel the mass (m) from both sides of the equation since it is the same on both sides: gh = 0.5 * v^2.
Step 9: Rearrange the equation to solve for v^2: v^2 = 2gh.
Step 10: Take the square root of both sides to find the final velocity: v = √(2gh).