Suppose a hockey puck slides down a frictionless ramp with an acceleration of 4.70 m/s2. What angle does the ramp make with respect to the horizontal?
Please show steps...
To find the angle that the ramp makes with respect to the horizontal, we can use trigonometry. Specifically, we can use the inverse tangent function (tan⁻¹). Here are the steps to find the angle:
1. Determine the vertical acceleration (acceleration in the direction perpendicular to the plane of the ramp).
- In this case, the vertical acceleration is equal to the acceleration of the puck because there is no friction to cause any additional vertical acceleration.
2. Use the formula for vertical acceleration to find the vertical component of the acceleration.
- The formula for vertical acceleration is given as a = gsinθ, where "a" is the vertical acceleration, "g" is the acceleration due to gravity (9.8 m/s²), and "θ" is the angle of the ramp.
- Since the ramp's vertical acceleration is equal to the acceleration of the puck, we can write the equation as 4.70 m/s² = 9.8 m/s² * sinθ.
3. Rearrange the equation to solve for sinθ.
- Divide both sides of the equation by 9.8 m/s²: sinθ = 4.70 m/s² / 9.8 m/s².
4. Use the inverse tangent function (tan⁻¹) to find the angle θ.
- Take the inverse tangent of both sides of the equation: θ = tan⁻¹(4.70 m/s² / 9.8 m/s²).
5. Calculate the angle using a calculator or math software.
- Plug in the value of (4.70 m/s² / 9.8 m/s²) into the inverse tangent function on a calculator: θ ≈ 27.7°.
Therefore, the ramp makes an angle of approximately 27.7° with respect to the horizontal.