Calculating Sled Speed on a Frictionless Hill: Equations for Theta and Radius

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In summary, the conversation involves finding the equation for calculating the speed of a sled starting from rest at the top of a frictionless hemispherical hill, given the angle and radius. The suggested approach is to use the conservation of mechanical energy and Newton's Second Law to find the final speed.
  • #1
mookie84
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Homework Statement


A sled starts from rest at the top of the frictionless, hemispherical hill. What is the equation to calculate the sleds speed at angle (theta)

Homework Equations





The Attempt at a Solution



I need to figure out the equation. I'm given theta and the radius. I thought the equation was Rsin(theta) but that was incorrect
 
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  • #2
We're told that the hill is frictionless, so we can use the conservation of mechanical energy to find the sleds speed at some angle theta. After finding the final speed, you'll want to use Newton's Second Law along with the equation for centripetal acceleration to find the sleds speed at some angle theta. Can you show me how you would set up the equation for the conservation of mechanical energy?
 
  • #3
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The equation to calculate the sled's speed at a given angle (theta) on a frictionless, hemispherical hill is v = √(gR(1-cos(theta))), where v is the sled's speed, g is the acceleration due to gravity, and R is the radius of the hill. This equation is derived from the conservation of energy principle, where the initial potential energy at the top of the hill is converted entirely into kinetic energy at the bottom of the hill. The radius of the hill is also taken into account because it affects the distance the sled travels while descending the hill.
 

1. How do you calculate the speed of a sled on a frictionless hill?

To calculate the speed of a sled on a frictionless hill, you will need to use the equation v = √(rgsinθ), where v is the speed, r is the radius of the hill, g is the acceleration due to gravity, and θ is the angle of the hill.

2. What is the significance of theta and radius in the equation for calculating sled speed?

Theta and radius are both important factors in determining the speed of a sled on a frictionless hill. Theta represents the angle of the hill, which affects the force of gravity and therefore the acceleration of the sled. The radius of the hill determines the distance traveled by the sled, which also affects its speed.

3. Can this equation be used for any slope or does it only work for frictionless hills?

This equation is specifically designed for calculating sled speed on frictionless hills. If there is friction present, the speed of the sled will be affected and a different equation will need to be used.

4. How does the mass of the sled affect the speed on a frictionless hill?

The mass of the sled does not affect the speed on a frictionless hill, as the only forces acting on the sled are gravity and normal force (which is equal to the sled's weight). Therefore, the mass will cancel out in the equation v = √(rgsinθ).

5. Is there a maximum speed that a sled can reach on a frictionless hill?

According to the equation v = √(rgsinθ), there is no maximum speed that a sled can reach on a frictionless hill. However, the angle of the hill will affect the speed, as a steeper slope will result in a higher speed. It is also important to consider any potential obstacles or limitations that may apply in a real-life scenario.

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