Final answer:
The minimum speed a 0.02 kg puck needs to reach the top of a frictionless ramp inclined at 30° is 9.8 m/s, calculated using the conservation of energy principle.
Step-by-step explanation:
To find the minimum speed that a 0.02 kg puck needs to reach the top of a frictionless ramp inclined at 30°, we can utilize the principles of conservation of energy. Since there are no non-conservative forces doing work (such as friction or air resistance), mechanical energy is conserved. Therefore, we equate the kinetic energy at the bottom of the ramp to the potential energy at the top of the ramp:
Kinetic Energy at bottom = Potential Energy at top
(1/2)mv² = mgh
Where m is the mass of the puck, v is the velocity, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the ramp. Since the ramp is inclined at 30°, we can calculate h as the opposite side of a right triangle (h = L * sin(30°)) where L is the length of the ramp (9.8 m).
Solving for v:
v² = 2gh
v = √(2 * 9.8 m/s² * 9.8 m * sin(30°))
v = √(2 * 9.8 m/s² * 9.8 m * 0.5)
v = √(2 * 9.8 m/s² * 4.9 m)
v = √(96.04 m²/s²)
v = 9.8 m/s
Therefore, the minimum speed the puck needs to make it to the top of the ramp is 9.8 m/s.