What is the acceleration of patrons when the speed doubles and the radius also doubles during the ride?

To determine the acceleration of patrons when the speed doubles and the radius also doubles, it's important to understand how centripetal acceleration works in circular motion. The formula for centripetal acceleration (a_c) is given by:

[ a_c = \frac{v^2}{r} ]

where ( v ) is the linear speed and ( r ) is the radius of the circular path.

When the speed doubles, it becomes ( 2v ). The new centripetal acceleration can be computed as follows:

[ a_c' = \frac{(2v)^2}{2r} ]

Calculating this step-by-step:

1. The speed squared when doubled: ( (2v)^2 = 4v^2 ).

2. The radius being doubled means we substitute ( r ) with ( 2r ) in the equation, giving us:

[ a_c' = \frac{4v^2}{2r} = \frac{4}{2} \cdot \frac{v^2}{r} = 2 \cdot \frac{v^2}{r} ]

This indicates that the new acceleration is essentially twice the original centripetal acceleration