What minimum speed does a 150 g puck need to make it to the top of a 5.90 m-long, 21.0 degree frictionless ramp?
To find the minimum speed required for the puck to reach the top of the ramp, you can use the principles of mechanical energy conservation. Here's how you can solve it:
1. Identify the known values:
- Mass of the puck, m = 150 g = 0.15 kg.
- Length of the ramp, L = 5.90 m.
- Angle of inclination, θ = 21.0 degrees.
2. Determine the height of the ramp:
- The height of the ramp, h, can be calculated using the formula h = L * sin(θ).
- Substitute the given values: h = 5.90 m * sin(21.0 degrees).
3. Calculate the minimum potential energy required:
- The minimum potential energy required for the puck to reach the top is given by m * g * h, where g is the acceleration due to gravity (9.8 m/s²).
- Substitute the values: Potential energy = 0.15 kg * 9.8 m/s² * h.
4. Convert the potential energy to the minimum kinetic energy:
- Since the puck needs to reach the top, the initial kinetic energy will be zero. So the minimum speed required at the bottom of the ramp is equal to the square root of twice the potential energy, according to the principle of conservation of mechanical energy.
- Kinetic energy = 0.5 * m * v^2, where v is the velocity of the puck.
- Set the potential energy equal to the kinetic energy: 0.5 * m * v^2 = Potential energy.
5. Plug in the values and solve for v:
- Substitute the known values and solve for v: 0.5 * 0.15 kg * v^2 = 0.15 kg * 9.8 m/s² * h.
- Simplify the equation and solve for v: v^2 = 9.8 m/s² * h.
6. Calculate v by taking the square root of both sides:
- Take the square root of both sides to find v: v = √(9.8 m/s² * h).
7. Plug in the value of h from step 2 and calculate v:
- Substitute the calculated value of h and solve for v: v = √(9.8 m/s² * h).
By following these steps, you will obtain the minimum speed (velocity) required for the 150 g puck to reach the top of the 5.90m, 21.0° frictionless ramp.