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A roller coaster is designed such that r...

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is `20m`. The speed of the car at the top of the hill is between

A

`14 m//s` and `15m//s`

B

`15m//s` and `16m//s`

C

`16m//s` and `17m//s`

D

`13 m//s` and `14m//s`

Text Solution

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The correct Answer is:
To solve the problem of determining the speed of a roller coaster at the top of a hill where riders experience weightlessness, we can follow these steps: ### Step 1: Understand the Forces Acting on the Rider At the top of the hill, two main forces act on the rider: - The gravitational force (weight) acting downward, which is \( mg \). - The normal force acting upward, which is \( N \). ### Step 2: Set Up the Equation for Weightlessness For the rider to experience weightlessness, the normal force \( N \) must be zero. Therefore, the gravitational force must provide the necessary centripetal force to keep the rider moving in a circular path. This gives us the equation: \[ mg - N = \frac{mv^2}{r} \] Since \( N = 0 \) for weightlessness, we can simplify the equation to: \[ mg = \frac{mv^2}{r} \] ### Step 3: Cancel the Mass \( m \) Since the mass \( m \) appears on both sides of the equation, we can cancel it out: \[ g = \frac{v^2}{r} \] ### Step 4: Rearrange to Solve for Speed \( v \) Rearranging the equation gives us: \[ v^2 = gr \] Taking the square root of both sides, we find: \[ v = \sqrt{gr} \] ### Step 5: Substitute the Given Values We are given: - The radius of curvature \( r = 20 \, \text{m} \) - The acceleration due to gravity \( g \approx 9.8 \, \text{m/s}^2 \) (or we can use \( g \approx 10 \, \text{m/s}^2 \) for simplicity) Using \( g = 10 \, \text{m/s}^2 \): \[ v = \sqrt{10 \times 20} = \sqrt{200} \] ### Step 6: Calculate the Speed Calculating \( \sqrt{200} \): \[ \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \approx 14.14 \, \text{m/s} \] ### Step 7: Determine the Range of Speed Since the question asks for the speed range, we can conclude that the speed of the roller coaster at the top of the hill is approximately between 14 m/s and 15 m/s. ### Final Answer The speed of the car at the top of the hill is between **14 m/s and 15 m/s**. ---
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