Question

In an amusement park ride, a rider suspended by cables swings back and forth from a tower. The maximum speed \(v\) (in meters per second) of the rider can be approximated by \(v=\sqrt{2 g h}\), where \(h\) is the height (in meters) at the top of each swing and \(g\) is the acceleration due to gravity \(\left(g \approx 9.8 \mathrm{~m} / \mathrm{sec}^2\right)\). Determine the height at the top of the swing of a rider whose maximum speed is 15 meters per second.

Short answer

The height at the top of the swing is approximately 11.47 meters.

Step-by-step solution

Identify the Given Variables

Identify the given quantities in the problem. Here we have two known variables: the maximum speed `v` which is given as 15 meters per second and the acceleration due to gravity `g` which is approximately 9.8 m/s^2.

Rearrange the Formula to Solve for `h`

The formula \(v=\sqrt{2 g h}\) needs to be rearranged to solve for `h`. This can be done by squaring both sides, which produces \(v^2= 2 g h\), and then dividing by \(2g\) on each side, which gives \(h = v^2 / (2g)\).

Substitute the Given Values into the Formula and Solve

Substitute the values of `v` (15 m/s) and `g` (9.8 m/s^2) given in the problem into the formula from step 2 to determine the value of `h`. Thus, \(h = (15)^2 / (2*9.8) \approx 11.47\) meters. Therefore, the rider's maximum speed of 15 meters per second is achieved at a height of approximately 11.47 meters.

Key concepts

Maximize Speed Calculation

Understanding how to calculate the maximum speed of an amusement park ride is fundamental to ensuring the safety and enjoyment of riders. The speed at which a rider travels can be directly tied to the exhilarating experience they have. When a rider reaches the peak of their swing in the ride, they momentarily come to a halt before descending back down, which is when they gain speed due to gravity.

Maximizing the speed can be understood by using the equation provided in our exercise, \(v=\sqrt{2gh}\), where \(v\) is the speed, \(g\) is the acceleration due to gravity, and \(h\) is the height. This equation is derived from the principles of energy conservation, specifically the conversion between potential and kinetic energy. At the top of the swing, the energy is mostly gravitational potential energy, which is maximized, and as the rider falls, this energy is converted into kinetic energy, which equals to the speed of the rider. To calculate the maximum speed at the bottom of the swing, one can rearrange this relationship into the provided formula.

In an amusement park setting, factors like friction and air resistance are often minimized to ensure that the ride can reach a greater speed. However, these factors can still affect the ride's speed, and subtle changes to the height \(h\) of the swing can have a significant impact on the maximum speed that the rider experiences.

Acceleration Due to Gravity

Acceleration due to gravity is a pivotal concept that affects many aspects of physics, especially when dealing with motion and forces on Earth. This acceleration, denoted by \(g\), is approximately \(9.8 \mathrm{~m/s^2}\) near the surface of the Earth. This value represents the rate at which an object accelerates when falling freely under the influence of Earth's gravitational pull.

Gravity's role in amusement park rides can't be overstated. For the thrilling experience of a swing ride, gravity is what pulls the rider back toward the ground after reaching the top of the swing. Not only does gravity determine the speed of the rider at the lowest point of the swing, but it also is a constant factor in all phases of the ride's motion.

Constant Across All Locations

Regardless of where you are on Earth, gravity acts downwards toward the center of the Earth with a nearly consistent acceleration. This constant provides a stable frame of reference for calculating various types of motion and for ensuring safety standards for amusement park rides. Gravity is why we don't float off into space and why a pendulum will eventually come to rest.

Kinematics Equations

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. Kinematic equations allow us to predict the future position, speed, and acceleration of an object moving under constant acceleration, like an amusement park ride influenced by gravity.

The kinematic equation applied in the step-by-step solution of our exercise is a simplification, specifically adapted for the scenario where the only force doing work is gravity. In more complex scenarios, other kinematic equations take into account initial velocity, final velocity, and time traveled.

Applicability in Real-World Scenarios

Real-world applications of kinematics extend beyond amusement park rides to include fields like robotics, ballistics, and even planetary motion. In each of these fields, understanding the interplay between displacement, velocity, and acceleration is critical for accurate predictions and designs. In the context of our problem, kinematics helps us understand the relationship between the drop height of the ride and the resultant maximum speed of the rider.