Final answer:
To calculate the minimum speed v for a ball to reach the top of a vertical tube from a height h, use the conservation of energy equation and solve for v = √(2gh), where g is the acceleration due to gravity. The tube's radius r does not affect this calculation.
Step-by-step explanation:
To determine the minimum speed v with which a small ball must be pushed inside a smooth vertical tube to reach the top, we can use principles of energy conservation. The ball needs to have enough kinetic energy to rise to a height h against the pull of gravity. If air resistance is negligible, the minimum speed can be calculated using the energy conservation equation:
KE_initial + PE_initial = KE_final + PE_final
Here, KE stands for kinetic energy, and PE for potential energy. At the height h, the kinetic energy will be zero if the ball just reaches the top without extra speed. Assuming the ball starts from rest at the bottom, the initial potential energy is zero. Thus, the equation simplifies to:
0.5 * m * v^2 = m * g * h
Cancelling out the mass m from both sides gives:
v^2 = 2 * g * h
By taking the square root, we get the minimum speed v:
v = √(2 * g * h)
Where g is the acceleration due to gravity. This relationship is independent of the radius r of the tube, and it's a result directly derived from the conservation of mechanical energy principle or Torricelli's theorem in the context of fluid dynamics.