What minimum speed does a 150g puck need to make it to the top of a 4.5m -long, 21degree frictionless ramp?

To determine the minimum speed required for the 150g puck to reach the top of the ramp, we can use the principles of energy conservation.

First, let's break down the problem. We have a frictionless ramp with a length of 4.5m and an angle of 21 degrees. The puck has a mass of 150g, which is equivalent to 0.15kg.

The key concept here is that the total mechanical energy of the puck remains constant. At the top of the ramp, all of the puck's initial kinetic energy will be transformed into gravitational potential energy.

Now, let's calculate the minimum speed required using the following steps:

1. Convert the mass of the puck from grams to kilograms: 150g ÷ 1000 = 0.15kg.

2. Determine the gravitational potential energy (PE) at the top of the ramp using the formula:
PE = m * g * h
where m is the mass of the puck, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height.

Since the ramp is at an angle of 21 degrees, the vertical height is given by:
h = 4.5m * sin(21°)

Substituting the known values:
h = 4.5m * sin(21°) ≈ 1.63m

PE = 0.15kg * 9.8m/s² * 1.63m ≈ 2.4 Joules

3. Equate the final gravitational potential energy to the initial kinetic energy (KE) to find the minimum speed. The initial kinetic energy is given by:
KE = 1/2 * m * v²
where v is the velocity/speed.

Setting the PE equal to KE:
2.4 Joules = 1/2 * 0.15kg * v²

4. Solve for v:
Multiply both sides of the equation by 2 to eliminate the fraction:
4.8 Joules = 0.3kg * v²

Divide both sides by 0.3kg:
v² = 4.8 Joules / 0.3kg

Simplify:
v² ≈ 16 m²/s²

Take the square root of both sides to find the minimum speed:
v ≈ √16 m²/s² ≈ 4 m/s

Therefore, the minimum speed the 150g puck needs to reach the top of the 4.5m-long, 21-degree frictionless ramp is approximately 4 m/s.