find the minimum speed for the roller coaster riding along the inner side of a loop with a radius r and an acceleration of 1.5 times earth's little g for each of the following for each of the following rollercoaster constructions:
a.the radius of curvature at the top of the loop is 15 m.
b.the radius of curvature at the top of the loop is 25 m.
To find the minimum speed for the roller coaster riding along the inner side of a loop, we need to use the concept of centripetal acceleration.
The centripetal acceleration is given by the formula:
a = v^2 / r
Here, a represents the centripetal acceleration, v represents the velocity of the roller coaster, and r represents the radius of curvature at the top of the loop.
We are given that the acceleration of the roller coaster is 1.5 times Earth's little g, where g is the acceleration due to gravity on Earth. Let's assume the value of g is 9.8 m/s^2.
a = 1.5g = 1.5 * 9.8 = 14.7 m/s^2
Now, let's calculate the minimum speed for each of the given roller coaster constructions:
a) For a radius of curvature of 15 m:
We know that the acceleration a is equal to v^2 / r.
Substituting the values, we have 14.7 = v^2 / 15.
To find the minimum speed, we need to solve for v. Rearranging the equation, we get:
v^2 = 14.7 * 15
v^2 = 220.5
v ≈ sqrt(220.5)
v ≈ 14.85 m/s (rounded to two decimal places)
The minimum speed for the roller coaster with a radius of curvature of 15 m is approximately 14.85 m/s.
b) For a radius of curvature of 25 m:
Using the same formula, we have 14.7 = v^2 / 25.
Solving for v, we get:
v^2 = 14.7 * 25
v^2 = 367.5
v ≈ sqrt(367.5)
v ≈ 19.17 m/s (rounded to two decimal places)
The minimum speed for the roller coaster with a radius of curvature of 25 m is approximately 19.17 m/s.