Vectors and Classical Mechanics

Introduction

Classical mechanics is the branch of physics that studies how bodies move, and our understanding helps us analyze and predict the movement of objects on Earth and in our solar system, galaxy, and the universe.  The motion of an object requires that we know its position at any given point in time, which means vectors are essential for mathematically calculating movement since direction and magnitude are vital to know.

Conceptually Understanding Movement

Although there are mathematical equations and calculations that follow, conceptually understanding how and why things move are critical to apply physics correctly.  We will be using the free simulations and virtual experiments at Advanced Tools for e-Learning to explore the physics of moving objects.

We will be building from the following:

• fundamentals of vectors (magnitude, direction, and axial components) and
• vector addition.

Kinematics

Kinematics studies the patterns of how things move.  The following variables create the mathematical structure for describing motion:

• Position is the place where something is located within a mapped space, and for the following, we’ll label it as x.
• Displacement (△x) is the change in position of an object:  

△x = x₁ – x₀     

Note that Δx may be positive or negative, which indicates direction relative to the x-axis.

• Velocity (v) describes how displacement changes with time:  

v = △x / △t  = (x₁ – x₀) _/ (t₁ – t₀)

Where △t is the change in time that the motion was observed, and the subscripts denote when and where the object was observed.

• Acceleration measures how velocity is changing over time:  

a = △v / △t = (v₁ – v₀) _/ (t₁ – t₀)

These relationships lead to four key kinematic equations for objects moving at constant acceleration:

△x = (v_i + v_f) / 2 * t

△x = v_i * t + 0.5 * a * t²

v_f = v_i + a * t

v_f² = v_i² + 2 * a * △x

Where t is the time since the observing the motion began,  v_i the initial velocity, and v_f the final velocity (or that observed at time t).

Explore how these equations are derived from the velocity vs. time graph.

Dynamics

Dynamics focuses on why things move, and this means we study forces that create a change in motion, which is also known as acceleration. 

The acceleration of an object is equal to the force acting on an object divided by the mass of the object, or rewritten as F = ma.

Since the kinematic equations above describe the motion of objects moving with constant acceleration, this means that the force(s) acting on the object must remain constant as well.  Objects moving with constant acceleration are the simplest to work with, helping us to understand the more complex movement of water and air on Earth, which are not moving at a constant acceleration.

Applying Classical Mechanics to Golf

Using Resources at Advanced Tools for e-Learning (ATeL)

ATeL has created a folder of their free activities that support Science Pickle*, which includes a variety of activities to study one- and two-dimensional motion in the context of the game of golf:

1. Simulations let you explore how the many moving parts of a complex process fit together.  In these simulations, explore how position, velocity, acceleration, and forces are involved in putting (straight-line motion in one-dimension when putting horizontally, and two-dimensional motion when the green is slanted) and chipping a golf ball (two-dimensional motion through the air).
2. Virtual experiments guide you through an investigation.
3. Lessons that define and illustrate concepts as needed.
4. Assessments allow you to check your knowledge and understanding.
5. A wizard-driven authoring tool enables you to modify existing virtual experiments and assessments and create new ones.  Teachers may design activities tailored for their classes, and learners may use the tool to prepare for a test.

^* There are many more topics available to explore at ATeL’s website.  Register to get access to their free activities.

Force of Gravity (F_g) or Weight

Bodies of mass gravitationally attract each other with force proportional to the mass of each object and inversely proportional to the square of the distance between their centers.

Force of gravity = F_g = G * m₁ * m₂ / d²

where G = gravitational constant,

m₁ = mass of the first object,

m₂ = mass of the second object, and

d = distance between the centers of the two objects.

If a body is on the Earth’s surface (or relatively close to it), then let m₁ = mass of the Earth (m_Earth), m₂ = mass of the body of interest, and d = radius of the Earth (Radius_Earth).

Since F = m * a, then

F_g = m * a_g = m * G * m_Earth / Radius_Earth²

Therefore, a_g = G * m_Earth / Radius_Earth² = 9.8 m/s² = “g” in most textbooks.

The force of gravity acting on a body at or near the Earth’s surface is commonly called the object’s weight, which we’ve just shown is

F_g = weight = m * g

The weight of the golf ball points toward the center of the Earth.

CAUTION: It is essential to recognize when “m” is referring to mass or a unit of length when working in physics (or chemistry).

Normal Force (F_N)

The normal force is the component of a contact force that is always perpendicular to the surface that an object contacts.  Both objects may be solids, but they don’t have to be.  In this case, the solid ground pushes perpendicular against the surface of contact with the golf ball. If the green is horizontal, the atoms that make up the ground create a barrier for the ball to move due to the force of gravity.

Note: the term “normal” has many meanings.  In this usage, “normal” means “perpendicular.”  The contact force between the two objects is perpendicular to the surface of contact.

Friction (F_fr)

Friction is the component of a contact force that is always parallel to the surface of contact.  Friction opposes motion.  If the object is not moving, static friction opposes any force trying to push it along the surface.  If the body is sliding, kinetic friction acts in the direction opposing the motion.  In most cases, friction is independent of the contact’s surface area and is proportional to the normal force. 

Force of friction = F_fr = μ * F_N

where μ is the coefficient of friction.

The coefficient of friction depends on the roughness of both surfaces that are in contact.  When both bodies are stationary to each other, μ is called the coefficient of static friction (μ_s).  When one body is sliding along the surface of the other, μ is called the coefficient of kinetic friction (μ_k), and the value is independent of the speed of the sliding.

Note: μ_k ≤ μ_s

The coefficient of friction of golf greens changes throughout the day as the grass grows and as the amount of moisture in the blades dries in the intense afternoon sun. 

Air Resistance (F_ar)

Air resistance is similar to friction in that it always opposes the motion of an object.  But unlike friction, its magnitude depends on the speed of the object moving through the air. The faster an object moves, the higher the air resistance.  The magnitude also depends on the cross-sectional area relative to the motion.  A skydiver slows down by spreading themselves out perpendicular to their motion, and speed up by falling head first.

Putting: Straight-Line Motion

Forces Acting on a Putted Golf Ball

This simulation does not deal with how the golf ball got moving. Instead, it is about what happens to the ball once it leaves the putter and moves across the green.  Three forces act on the moving ball:

1. the force of gravity (mg) pulling the ball toward the center of the Earth,
2. the normal force (F_N) of the ground (green hashed surface) pushing perpendicularly on the ball, and
3. the force of friction (F_fr) between the surface of the golf ball and the land.  Friction always acts parallel to the surface of contact and opposes the direction of motion.

If the green is slanted, the weight of the ball has components of force parallel and perpendicular to the surface.

An illustration of the forces acting on a golf ball once putted on a horizontal surface (hashed green lines).  Since the ball is not leaving the surface, the normal force must equal the weight of the ball so the forces are balanced in the vertical direction.  Since there is only one force acting in the horizontal direction, the ball must be accelerating in the direction of the friction, which is always opposite the motion of the ball.

Simulations may be used to run experiments as if you had the equipment and were on a golf course.  Try the following:

Chipping: Motion in Two Dimensions 

Forces Acting on a Golf Ball Moving Through the Air

Unlike putting on a flat green, a golf ball moving through the air changes direction.  Breaking up the movement into vertical and horizontal components allows us to apply the kinematic equations in each direction.

The force of gravity is always acting vertically downward on the ball as it moves through the air.  If the ball is moving upward, gravity is slowing its rate of ascent, but once it begins to fall back to Earth, the ball accelerates downward.

When there is air resistance, the acceleration of the golf ball is not constant because the forces acting on the ball are continually changing.   Air resistance always opposes the motion of the golf ball, and its magnitude depends on the velocity of the ball.