Chapter 1: Newton's Laws: Motion and Forces

Motion and forces are known to be challenging concepts. We experience the relation between force and motion all the time in our everyday lives, yet we are often unaware of our motion in space. This book offers a number of examples and investigations to complement traditional physics textbook presentations of motion and increase awareness. The representations include the experience of the body and experiments with simple toys as well as photos, video analysis, and electronic data taking in authentic situations in amusement parks (Fig. 1.1). These experiences can be described as distilled “everyday physics”: Where else can you move in perfect circles or fall freely for a couple of seconds and live to tell the tale? The idealizations in these out-of-school physics activities offer teaching opportunities that can be adapted to many degrees of difficulty, from preschool to university.

Newton’s laws also relate force and motion in playgrounds and on amusement rides. Your own body experiences the forces required for the different motions in ways that can be very concrete but also quite challenging (Fig. 1.1). Before embarking on exciting examples, we recapitulate Newton’s laws and then discuss briefly how smartphones can be used for measurements on amusement rides.

Earth moves nearly 30 km/s or 108 000 km/h in its orbit around the sun, whereas the fastest roller coaster in the world reaches a speed of 240 km/h. Why do we not feel the motion of the earth, when riding in a large roller coaster can make us lift from the seat or feel five times heavier than normal? When do we feel the motion of a train or car or airplane? The forces acting on us are related to Newton’s laws, summarized in this chapter.

If a body is at rest, you know that all forces acting on that body cancel. Everything on Earth is pulled down by the force of gravity, which has to be counteracted by other forces for a body to remain at rest. When gravity pulls a book on a table, the table prevents the book from falling down by exerting an equally large normal force in the opposite direction. The upward force from the floor prevents the table from going through the floor. When you stand in a queue waiting to enter an amusement park, the ground pushes you up just as much as gravity pulls you down. The book, the table, and you remain at rest.

The force of gravity increases with your mass m and can be written as mg, where g is the acceleration of gravity, which has a value g ≈ 9.8 m/s² and is directed towards the center of the earth. The bold face in g denotes that it is a “vector,” which has both size and direction. The “normal force,” N, is orthogonal to the contact area. The vector notation gives a compact way to describe that the normal force counteracts gravity: N = −mg. The forces can also be illustrated with arrows in a “free-body diagram” as shown in Fig. 1.2.

Newton’s first law—the law of inertia—states that “a body remains at rest, or in uniform rectilinear motion, unless influenced by unbalanced forces.”

Newton’s first law thus includes the remarkable observation that no force is needed to maintain motion with constant velocity. We do not feel motion! Have you ever been on a train in a railway station, seeing a neighboring train move slowly and been unsure about whether your train has started moving? The law of inertia applies as you move slowly up to the top of a drop tower or up a roller coaster, as in Fig. 1.1. In all these cases, the velocity is constant and the sum of all forces acting on you must be zero.

A group of eleven-year-olds described their experience of inertia using their own words relating to a ride in bumper cars (Fig. 1.3). When you collide in a bumper car your body wants to continue moving forward. The body has got used to the speed. The concept of inertia, as expressed by Newton’s first law, seemingly contradicts our everyday experiences. For example, you need to keep pedaling your bike just to maintain speed—but you are also exposed to forces opposing your motion, e.g., from the air and from tires getting deformed.

Newton’s first law also describes what is required to change a motion with constant velocity: A force.

Nobody visits an amusement park to experience Newton’s first law, but rather to experience acceleration—the change in motion resulting from unbalanced forces in launches, drops, twists, turns, and stops. The relation between forces and acceleration are described quantitatively in Newton’s second law.

Newton’s second law of motion gives a mathematical relation between the force, the inertia, and the rate of change of the velocity, i.e., the acceleration a. It is traditionally written as

The larger the mass m, the larger the inertia and the larger the force required to accelerate the mass. In physics, the term acceleration describes change in velocity—not only increase of speed, but also decrease of speed, as well as changes in direction. In fact, the largest accelerations on amusement rides are usually changes of direction, such as when you pass the lowest point of a large swing or roller coaster valley, or as a roller coaster makes a sharp hairpin turn before reaching the station.

Newton’s second law is closely related to the experiences of the body. During an acceleration, a, every cell in the body must experience a force, F, required to accelerate its mass, m, as highlighted by rewriting Eq. (1.1) as an expression for acceleration:

Defined through Newton’s second law in this form, acceleration is no longer an abstract mathematical concept, but a very concrete experience throughout the body that is accelerating. In this approach, the concept is also accessible to young learners long before kinematic relations.

Newton’s third law emphasizes interaction: Every force is the result of an interaction between two bodies. When a force due to object B acts on object A, then an equal and opposite force due to object A acts on object B.

To understand the forces you experience as you are accelerating on amusement park rides, it is important that you focus on the forces acting on you—not on the forces you exert on the surroundings. It is also important to remember that a force acting on your body must originate from an interaction with another body.

When your bumper car collides head on with another car, the car pushes your body backwards to stop your forward motion—and you push the car forward. As a car turns, it has to push your body to make it move along in the turn—otherwise your body would continue straight ahead. When the car pushes you, you push back on the car with a force with the same size but opposite direction, according to Newton’s third law, the law of action and reaction.

In Secs. 1.2–1.4, we introduced Newton’s three laws of motion:

1. The law of inertia: A body remains at rest, or in uniform rectilinear motion, unless influenced by unbalanced forces

2. The law of acceleration: a = F/m

3. The law of action and reaction: When a force due to object B acts on object A, then an equal and opposite force due to object A acts on object B

These laws describe the relations between force and motions in the solar system and in our everyday lives, and they give us tools to understand the forces acting on our bodies as we experience them on amusement park rides, for example.

Although the laws can be stated in very compact form and are easy to write down, their consequences are rich and sometimes surprising. It is not enough to learn these laws by heart—applying them requires practice in a variety of contexts. The conceptual understanding can be quite demanding.

Student problems relating to force and motion are well-documented in physics education research and also prove surprisingly resistant to teaching, as reviewed, e.g., by McDermott and Redish (1999) and Hake (1998). The list that follows shows a few consequences of Newton’s laws that are relevant for studying forces on amusement park rides, but often forgotten by students:

1. Constant velocity requires no force.

2. Velocity, acceleration, and forces are vectors, with both size and direction.

3. Moving in a circle involves a change of direction and thus involves acceleration.

4. Forces acting on your body must be distinguished from forces you exert on your surroundings.

This book includes a large number of examples to deal with these concepts to complement more traditional textbook approaches.

The introduction to the laws of motion can start early. For example, the document benchmark for science literacy (AAAS, 2006) suggests that students by the end of the 5th grade should know that “changes in speed or direction of motion are caused by forces” and also that “the greater the force is, the greater the change in motion will be. The more massive an object is, the less effect a given force will have.”

Newton’s laws talk about “bodies,” neither implying nor excluding human bodies. Forces acting on our own accelerating bodies need not be abstract. In addition to the experience of the body, the forces required for acceleration can be visualized with simple toys as measuring devices during the accelerated motion. Smartphone sensors can provide additional graphic representations. As the motions of our own bodies are rarely in only one dimension, it is also natural to introduce the concept of vectors from the start.

The first-person approach in this book differs from traditional textbook treatments of force and motion, which often start with extensive examples of non-motion, followed by kinematics, with mathematical descriptions of one-dimensional motion with constant velocity or uniform acceleration and with forces acting primarily on inanimate objects.

Velocity, acceleration, and force are vectors, which have both magnitude and direction. Where you end up depends on which direction you are moving. Vectors are often represented by arrows, but can also be expressed in terms of coordinates.

As long as the motion is only in one dimension, a sign is sufficient to define if you are moving right or left in horizontal motion or up or down for vertical motion. More general velocities in three dimensions can be described in terms of a coordinate system XYZ. The axes can be chosen in any way suitable for the situation studied, but a common choice is to let the Z-axis point up. The acceleration of gravity, g, then points in the negative Z direction. As you stand on the ground, the normal force, N, required to counteract the force of gravity, mg, points in the positive Z direction.

We use the convention that a vector is denoted with bold face but the magnitude of the vector with ordinary text. For example, the speed, v, which tells only how fast something moves but not in which direction, can be written as v = |v|.

Our own bodies are not point particles and the orientation of our body matters. Going up a roller coaster lift hill with constant velocity feels quite different from standing on the ground. Forces acting on your body are often expressed in a coordinate system that is reoriented with the body, with the x-axis pointing forward, the y-axis to the left, and the z-axis along your spine toward the head, as illustrated in Fig. 1.4. As you stand on the ground, the Z direction coincides with the positive z direction of your body.

In a roller coaster lift hill, the force from the roller coaster on your body needed to cancel the force of gravity comes from both the seat and the backrest, corresponding to the x- and z-axes of your body. Figure 1.5 shows a free-body diagram of the forces on a rider as the roller coaster train pulls up a 30° lift hill. This “normal force” from the ride preventing the rider from falling straight down can be divided into the x- and z-components in the coordinate system of the rider. The vector sum of these forces, N_x and N_z, balance exactly the force of gravity, mg, on the rider.

In Malmö central station images of landscapes moving past train windows are projected on the concrete walls. As you watch them you may get the impression that you are moving through the landscape. Sitting in a train, waiting for departure, you sometimes see a neighboring train moving slowly and are uncertain whether you have already started to move. As motion with constant velocity requires no additional force, you do not feel motion.

Similarly, amusement parks sometimes create an illusion of vertical motion. A bit of rattle to simulate the start of an elevator in a dark ride can be followed by images of motion through the window, making you believe that you have moved a long way up or deep down an elevator shaft. Velocity is relative—no forces are required to keep a constant velocity, and without a defined reference we can thus not discern if it is our bodies or the surroundings that are moving.

However, while velocity is relative, acceleration is absolute. From a purely mathematical point of view, it is, of course, possible to define a relative acceleration between two systems. You could mathematically describe a system at rest as accelerating away from you, even if you are accelerating. But Newton’s second law tells us that you can’t have acceleration without a force. When it is your body accelerating, you would feel that force throughout your body. The experience of the forces required for acceleration makes Newton’s second law more intuitive than the first.

Horizontal acceleration requires a horizontal force. This implies that the total force from the floor or seat will deviate from the direction of gravity. Tilting the floor or seat and surrounding you with images corresponding to accelerated motion, simulators, or indoor rides can trick your senses to believe that you are, indeed, speeding up or slowing down. A moderate horizontal acceleration has very little effect on the total force acting on you from the seat or floor. It increases with a factor $1+(a/g)2≈1+a2/2g2$⁠. Although acceleration is absolute in classical mechanics, it cannot be distinguished from gravity.

Bouncing on a trampoline during sunrise, you may see the sun when you are at the highest point, only to see it disappear behind trees as you are back on the trampoline mat, as seen in Fig. 1.6. As you move up and down, the sun also seems to move up and then down, over and over again. In what ways would the experience be different if you were standing still and instead viewing the changing scenery on a big screen or through virtual reality (VR) glasses? Remember that acceleration requires a force. The large acceleration as you change your direction of motion at the bottom of the bounce requires a large force acting on your body, making you feel momentarily much heavier than normal.

The photos in Fig. 1.6 include a spiral toy rabbit, where the length of the spiral is proportional to the force it exerts on the feet of the rabbit. (According to Newton’s third law, the feet exert a force on the spiral of the same size but opposite direction.) A bouncer who has left the trampoline mat is falling freely and the spiral is contracted, as no force from the spiral is needed to keep the rabbit feet moving together with the bouncer. The contracted spiral illustrates the weightlessness of free fall. When instead the bouncer turns around at the bottom of the motion, the spiral is considerably extended to exert sufficient force for the feet of the toy rabbit to accelerate upwards. The spiral toy is a dynamometer in disguise.

For moderate extensions, the tension force T exerted by the spiral on the rabbit’s feet follows Hooke’s law and is proportional to the extension. This gives T_z = −kz, where k is the “spring constant” of the spiral (Fig. 1.7), with T being directed upward when the feet are pulled down. Using Newton’s second law, this force on the feet can also be related to acceleration, giving ma_z = −kz − mg. At rest, where a = 0, the extension z₀ of a free-hanging spiral is given by kz₀ = −mg, giving z₀ = −mg/k. For an upward acceleration, like at the bottom of the bounce, the spiral extends well beyond the equilibrium length, as seen in Fig. 1.8.

Many electronic accelerometers include a small capacitor, where the distance between the plates changes with the acceleration, just as the distance between the coils in the slinky or spiral rabbit toy. The changing distance leads to a changed voltage and an electric signal as a measure of the acceleration. It should be noted that an electronic accelerometer, just as the spiral toy, indicates “zero g” for the acceleration of free fall and −g when at rest.

The moderate accelerations experienced in an elevator (lift) may be studied with a bathroom scale, as well as with an accelerometer and pressure sensor, as studied by Monteiro and Martí (2016). Figure 1.9 shows data from an elevator (lift) in a tall building, indicating that a bathroom scale would show 5% to 6% more as the elevator accelerates up than when at rest or moving with constant velocity. Similarly it would show 5% to 6% less than normal when it is accelerating downwards—including the slowing up at the top.

Forces felt by your body can be captured by accelerometers, e.g., in your smartphone moving together with you. The data collected can be used for analysis directly after an amusement park ride or saved for analysis back in a classroom.

The accelerometer sensor measures the force on a small test body—except for gravity. At rest, whatever is holding the test mass in the smartphone must compensate exactly for the downward force of gravity. Thus, accelerometers do not—in spite of the name—measure acceleration but instead the vector G = a − g [or possibly G/g = (a − g)/g, depending on the settings]. This holds true whether you use traditional accelerometers or the accelerometer in your smartphone.

Download an app to access the accelerometer sensor—see, for example, Vieyra and Vieyra (2014) and Staacks et al. (2018)—and start the accelerometer. At rest, where a = 0, it measures −g in its own coordinate system. Observe how the reading changes as you orient your phone in different directions, e.g., leaning it up as if it were travelling on a seat of a roller coaster moving up a lift hill. Independent of orientation, the values shown on the smartphone for the different coordinates give a vector sum which cancels the acceleration of gravity. Similarly for motion with constant velocity, the sensor measures −g in its own coordinate system. If you let the phone drop a short distance onto a soft surface, the accelerometer data will show zero for the duration of the fall.

If your phone has a rotation sensor, your app may have an option “without g,” which displays a. Try both options and compare the output as you tilt your phone to an angle from the table. The option “acceleration without g” involves a considerable amount of hidden math to perform the “dead reckoning” needed for “motion tracking.” It gives output that may more closely resemble what you see in a traditional physics textbook. However, this option fails to discern the forces acting on your body as you move up a roller coaster lift hill from when you are standing on the ground. Our bodies are not point particles and orientation matters.

Throughout this book we will show data for the G-force vector G = a − g, collected by a sensor moving together with you. These data describe the forces acting on your body: A first-person perspective on motion!

Inclined planes are popular textbook examples to illustrate vector addition of forces in different directions, as shown in Fig. 1.10, where the normal force N prevents the body from going through the plane and the forward force F_f keeps the body from accelerating down the slope. If the vector sum N + F_f + mg = 0, the body remains at rest or keeps moving with constant velocity. Traditional roller coasters start with the train moving up an inclined plane—the lift hill.

An introduction to combinations of forces along different smartphone coordinates can be obtained by using the Physics Toolbox Play app (Vieyra et al., 2020), for example, where the change in data can be observed as the smartphone is reoriented around different axes.

The handrail of an escalator can provide an inclined plane for a smartphone and the slope can be established through direct measurement of one of the steps (Fig. 1.11). Escalators connect different levels in stores, train stations, and amusement parks, such as those shown in Fig. 1.12. Most escalators also have the same slope (30°) as the Lisebergbanan lift hill in Fig. 1.5. Your own experience is, of course, very different: While a person stands up on an escalator, the rider in a roller coaster is tilted while on the lift hill, in the same way as the smartphone going up an escalator.

Escalators are more easily accessible than roller coaster lift hills and offer good practice in understanding sensor data, as discussed in Pendrill (2020). The accelerometer graphs from a smartphone riding on the handrail of an escalator (Fig. 1.13) look very similar to the lift hill part of accelerometer graphs from rides on traditional roller coasters.

As an escalator moves upward (or downward) after the first transition steps, the motion is an example of Newton’s first law, where the force from the escalator on a person riding it must compensate for the force of gravity. This also holds true for a smartphone placed on the handrail. However, whereas the force on a person points straight up from the step, the smartphone axes are aligned with the slope.

The normal force on the phone (defined as the z-axis in the graphs) is orthogonal to the handrail. For θ = 30° it should be $mgcosθ=mg3/2$⁠, while the force along the forward direction of motion (defined as the x-axis in the graphs) is given by mg sin θ = mg/2 on the way up (and the opposite sign going down). These values agree well with the data shown in Fig. 1.13.

Some telephones also include barometers, recording the variations in air pressure. This can be very helpful for the interpretation of the motion data, e.g., from a ride on a large roller coaster.

If your phone has a pressure sensor, it can show how the air pressure varies as you move up or down an escalator, as in the bottom graph of Fig. 1.13. This can be used to provide an internal consistency check of the data obtained from the smartphone, as shown in Fig. 1.13.

The pressure difference over the longest escalator shown in Fig. 1.13 is Δp ≈ 2 hPa. For an air density $ρ≈1.3kg/m3$ this gives Δh ≈ 16 m. A manual count in a non-moving escalator found 87 steps with 20 cm height, but less for a few steps at either end. Approximately 80 steps times 20 cm/step also gives Δh ≈ 16 m. The graph shows that the uphill ride in the longest escalator takes about 64 s, corresponding to a vertical velocity component of 0.25 m/s.

The international standard slope for long escalators is 30°, which was also obtained by measuring the height and length of the steps as shown in Fig. 1.10(b). The speed 0.5 m/s is found to “give optimal capacity for a continuous flow of people.” This speed thus corresponds to a 0.25 m/s vertical velocity component, consistent with the measurements presented here.

Throughout the book, authentic sensor data are used together with photographs, drawings, and video screen shots to illustrate the different concepts and examples. Combinations of representations have been found to be more effective if they include different design dimensions, as, e.g., concrete vs abstract or physical vs virtual in Rau (2017). Each visual representation also needs to provide information about the learning content that partially overlaps with information provided in the other visual representations, while also adding relevant information not fully covered by the other representations.

This book invokes the experience of the body in addition to equations and visual representations. Sensor data provide a bridge between a theoretical analysis and the experience of the body and enable more detailed comparisons between theory and experiment. Numeric data from the sensors can be used to give real-life examples of integration and differentiation, and simple equipment can give additional visual representations of the function of the sensors for acceleration (Chap. 2) and rotation (Chap. 4).

Later chapters discuss energy transformations, followed by examples of circular motion in horizontal and vertical planes, before moving on to pendulum rides and general three-dimensional motion in roller coasters and other aspects of amusement park and playground visits.

A few questions to ponder before moving on to the next section:

1. What is the maximum speed of your favorite roller coaster?

2. Why do we feel the motion in a roller coaster but not the motion around the sun or the center of the galaxy?

3. Figure 1.9 shows accelerometer and elevation data for a long elevator. Use the elevation data to draw a rough sketch of how the velocity varies with time.

4. Figure 1.14 shows more detailed acceleration graphs for the elevator shown in Fig. 1.9. Use the graph to estimate the maximum speed of the elevator. Every box contributes 0.2 g s ≈ 2 m/s.

5. What would the graphs look like as you move down an escalator instead of up as in Fig. 1.13?

6. If you have a phone with an accelerometer, what level of accelerations can your phone register if you shake it?

7. Count the rectangles over the angular velocity graph in Fig. 1.15. Can you convert your result to a change in angle?

1. Check the Roller Coaster Database (rcdb.com) for your favorite roller coaster. In 2021, the fastest roller coaster was Formula Rossa in Ferrari Land, Abu Dhabi, which reaches 240 km/h after a 4.9 s hydraulic launch. (What is the average acceleration during the launch?)

2. The velocity on a roller coaster changes all the time—we feel the acceleration (and the air moving past our faces). We don’t feel constant velocity (Newton’s first law).

3. See Fig. 1.16.

4. The area under the accelerometer graph in Fig. 1.14 includes approximately 15 boxes over the line for G = g. Each box leads to an upward velocity increase of about 2 m/s giving a total speed of 30 m/s. As the elevator comes to a stop at the top, the downward acceleration is slightly larger but during a slightly shorter time, again giving 15 “boxes” of slowing down. The speed extracted from the elevation graph, as shown in Fig. 1.16, is a few percent larger, about 33 m/s. (Noisy accelerometer data limit the precision.)

5. The force in the x direction will have the opposite sign as you move down the escalator. The air pressure will start at the lower level and increase as you come down. The rotation data will have opposite signs. Read more in Pendrill (2020).

6. Some phones can only register 2g in each direction, but many reach 4g or 8g.

7. Read more in Pendrill (2020).