Physics Education PAPER • OPEN ACCESS Velocity, acceleration, jerk, snap and vibration: forces in our bodies during a roller coaster ride To cite this article: Ann-Marie Pendrill and David Eager 2020 Phys. Educ. 55 065012 View the article online for updates and enhancements. This content was downloaded from IP address 138.25.78.25 on 01/02/2021 at 05:14 Phys. Educ. 55 (2020) 065012 (13pp) iopscience.org/ped Velocity, acceleration, jerk, snap and vibration: forces in our bodies during a roller coaster ride Ann-Marie Pendrill1 and David Eager2 1 National Resource Centre for Physics Education, Lund University, Box 118, SE221 00 Lund, Sweden 2 School of Mechanical and Mechatronic Engineering, University of Technology Sydney, PO Box 123 Broadway, Australia E-mail: [email protected] Abstract Changing acceleration and forces are part of the excitement of a roller coaster ride. According to Newton’s second law, F = ma, every part of our body must be exposed to a force to accelerate. Since our bodies are not symmetric, the direction of the force matters, and must be accounted for by ride designers. An additional complication is that not all parts of the body accelerate in the same way when the acceleration is changing, i.e. when there is jerk. Softer parts of the body provide varying levels of damping, and different parts of the body have different frequency responses and different resonance frequencies that should be avoided or reduced by the roller coaster designer. This paper discusses the effect of acceleration, jerk, snap and vibration on the experience and safety of roller coaster rides, using authentic data from a dive coaster as an example. Keywords: roller coasters, jerk, acceleration, vibrations, amusement park physics, snap 1. Introduction Acceleration and forces are at the heart of many amusement park experiences. Exposing our Original Content from this work may be used bodies to the changing forces triggers the fight or under the terms of the Creative Commons flight response, which in turn triggers our bodies Attribution 4.0 licence. Any further distribution of this work to release neuro-chemicals such as adrenaline and must maintain attribution to the author(s) and the title of the work, journal citation and DOI. dopamine [1], causing many people to consider 1361-6552/20/065012+13$33.00 1 © 2020 The Author(s). Published by IOP Publishing Ltd A Pendrill and D Eager amusement rides to be a lot of fun and an enjoy- 2. Theory, measurement and experiences able experience. of ride features According to Newton’s second law, accel- If you enjoy roller coasters, you will most prob- eration of a body with mass m requires a force ably know what to expect from the ride, just by = F ma. Since the force of gravity, mg, always looking at a photo, such as figure 1. The long acts, the additional force required from the ride straight launch to the left provides sufficient kin- ( − ) can be written as m a g , usually expressed in etic energy for the coaster train to pass the top. terms of a G force which can be seen as a force You may also recognise the light grey stators of = − per mass unit G a g. The G force is thus the linear synchronous motors (LSM) on the track. independent of mass and is often expressed in During the launch, your body will feel a force terms of the acceleration of gravity, g ≈ 9.8 m −2 from the back of the seat [6]. During the long s . It relates the force from the ride as it is in drop that follows, you will feel much lighter than motion to the force when it is at rest. When the normal, possibly even weightless as you move acceleration a is close to g, very little force is faster and faster. In the large valley that follows, required from the ride and you experience near- you will instead feel a few times heavier than weightlessness. Bouncing on a trampoline only normal. the force of gravity acts while you are in the air Roller coasters depend on the conversion of and you experience 0 g, but as the trampoline energy between potential and kinetic energy. Tra- bed enables the acceleration for the next bounce, ditional roller coasters start with a lift hill, bring- you may reach 7 g [2, 3]. Elite trampolinists may ing the train to the highest point of the ride, where reach even larger forces [3, 4]. This exceeds the it has maximum potential energy. As gravity pulls maximum G force of 6 g allowed by amusement the train down the first drop, the speed and kin- park standards. Unlike in an amusement ride, you etic energy increases and the potential energy may interrupt the bouncing motion at any time decreases until the train starts to move up again, on a trampoline by bending your legs and using gaining potential energy while losing kinetic them as shock absorbing pistons, should you feel energy. uncomfortable. Less traditional roller coasters are instead Roller coaster rides need to fall into the given initial kinetic energy through a launch, Goldilocks zone to be considered for amusement where the energy may be supplied by some parks. Too few g’s and the ride is boring. Too other means such as electromagnetic propulsion, many g’s and the ride is dangerous. Within this hydraulic launch, compressed air or a mechanical − Goldilocks zone there needs to be a range of spring which accelerate the train from 0 m s 1 rides that accommodate the varying needs of the to a maximum speed in a few seconds (see e.g. amusement park visitors. In rides for children, the [7]). Electromagnets can also be used for a launch : G-forces of a ride typically lie in the range 0 5 g– during the ride, as shown in figure 1. They can 2 g (e.g [5]). Family rides extend the range to also be used to regulate the speed and stop the 0 g–3 g. Thrill rides often involve 4 g or more, ride. as well as negative g’s, where the body lifts from Some energy losses due to air resistance and the seat, and must be held in place by the restraint rolling friction are unavoidable. However, these system. energy losses are surprisingly small and meaning- The time derivative of acceleration is referred ful estimates of the speed can be obtained from to as jerk. The higher derivatives are called elevation differences from the highest point or snap, crackle and pop [3], followed by the from a point with known speed, using the relation less known term lock. Just as we experience for free fall, giving acceleration differently to velocity, we experi- ence jerk differently to acceleration, snap dif- + 2 ≈ : ferently to jerk and so on for the higher deriv- 2gh v constant (1) atives. In this paper we will discuss jerk and snap and present some practical examples of The maximum speed after a first drop these. is always slightly smaller than expected from November 2020 2 Phys. Educ. 55 (2020) 065012 Velocity, acceleration, jerk, snap and vibration: forces in our bodies during a roller coaster ride Figure 1. The second electromagnetic (LSM) launch of the Helix roller coaster, followed by an inverted Top Hat and a large valley. equation (1) based on the elevation differ- ence. Similarly, launch coasters never reach 200 the maximum height calculated from the initial speed. Figure 2 shows the relation between maximum speed and maximum height for a number of roller coasters of different 150 heights. 100 2.1. Visual estimate of forces in a roller v (km/h) coaster valley Equation (1) can be used to express the speed v at the bottom of the valley in figure 1 in terms of 50 the speed v0 at the top and the elevation difference ∆H. For the estimates, we neglect energy losses and also contributions due to the finite length of 0 the roller coaster train [8, 9]. Using v > 0, this 0 20 40 60 80 100 120 140 0 h (m) gives an estimate of the upwards centripetal accel- eration ac at the bottom of the valley in terms of the radius of curvature, R: Figure 2. Relation between maximum speed and max- imum elevation difference for a number of roller coast- 2 ers. The solid line shows how the speed at free fall v 2 g∆H ≈ ac = > 3 g (2) depends on the distance fallen. The blue squares denote R R a few traditional roller coasters with lift hills, whereas where we have used the photo to obtain an approx- the red stars mark launch coasters. imation of the ratio ∆H/R ≈ 3/2, as indicated in figure 3. N = 4 mg, which is also found in the accelero- The normal force needs to compensate for the meter data [3, 6]. force of gravity and provide the force required An interesting aspect of the forces is that they for the centripetal acceleration. The force from do not depend on the scale, only the ratio between the ride on your body should then be close to elevations and radii (as long as we neglect energy November 2020 3 Phys. Educ. 55 (2020) 065012 A Pendrill and D Eager 4 3.5 3 2.5 2 1.5 1 0.5 0 57 58 59 60 61 62 63 64 (a) Helix, radius (b) Accelerometer data Figure 3. (a) Estimate of the relation between the elevation difference and the radius of curvature, resulting in an expected normal force of 4mg at the bottom of the valley.