To find the velocity of the stone at a height of 3.0 m above the ground, we can use the principle of conservation of mechanical energy. The initial potential energy at the starting height will be converted into kinetic energy at the desired height. We can equate the two energies to solve for the velocity.
The potential energy (PE) of an object at height h is given by the equation:
PE = mgh
where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
At a height of 5.0 m, the initial potential energy is:
PE_initial = m * g * h_initial = 2.0 kg * 9.8 m/s^2 * 5.0 m = 98 J
At a height of 3.0 m, the final potential energy is:
PE_final = m * g * h_final = 2.0 kg * 9.8 m/s^2 * 3.0 m = 58.8 J
Since mechanical energy is conserved, we can equate the initial potential energy to the final kinetic energy (KE) at a height of 3.0 m:
PE_initial = KE_final
98 J = (1/2) * m * v^2
Solving for v (velocity):
v^2 = (2 * PE_initial) / m v^2 = (2 * 98 J) / 2.0 kg v^2 = 98 J / 1.0 kg v^2 = 98 m^2/s^2
Taking the square root of both sides:
v = √(98 m^2/s^2) v ≈ 9.9 m/s
Therefore, the velocity of the stone at a height of 3.0 m above the ground is approximately 9.9 m/s.