Topic 05 – Linear Kinetics

Topic 05 – Linear Kinetics

Lesson Learning Outcomes

• Explain Newton’s three laws of motion
• Apply Newton’s 2^nd law of motion:
  • To determine the acceleration of an object if the forces action on the object are known
  • To determine the net force acting on an object if the acceleration of the object is known
• Define impulse and momentum
• Explain the relationship between:
  • Impulse and momentum
  • Mass and weight

Guiding Questions

• What are the Newton’s three laws of motion and how are they applied to analyze motion in sport and exercises?
• What are the differences and relationship between impulse and momentum?

Introduction

Dynamics is the branch of mechanics concerned with the mechanics of moving objects. Kinetics is the branch of dynamics concerned with the forces that causes motion. Newton’s 3 laws of motion and law of gravitation provides the basis for kinetics. Linear kinetics can explain the causes of linear motion.

Newton’s 1^st Law of Motion

Newton’s first law of motion states that every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. Simply stated – Newton’s first law of motion states that if no net external force acts on an object, that object will not move (it will remain in its state of rest) if it wasn’t moving to begin with, or it will continue moving at constant speed in a straight line (it will remain in its state of uniform motion in a straight line) if it was already moving.

Newton’s first law of motion provides the basis for the equations describing the horizontal motion of a projectile that we used in the previous topic. Newton’s first law of motion is also the basis for static equilibrium. Newton’s first law of motion can be expressed mathematically below:

Newton’s first law of motion may be interpreted in several ways:

1. If an object is at rest and the net external force acting on it is zero, the object must remain at rest.
2. If an object is in motion and the net external force acting on it is zero, the object must continue moving at constant velocity in a straight line.
3. If an object is at rest, the net external force acting on it must be zero.
4. If an object is in motion at constant velocity in a straight line, the net external force acting on it must be zero.

Newton’s first law of motion applies to the resultant motion of an object and to the components of this resultant motion.

Conservation of Momentum

Newton’s first law of motion provides the basis for the principle of conservation of momentum (if we consider only objects whose mass is constant). Linear momentum is the product of an object’s mass and its linear velocity. The faster an object moves, the more momentum it has. The larger a moving object’s mass, the more momentum it has. So, momentum is a way of quantifying the motion and inertia of an object together in one measure. Linear momentum if defined mathematically in the equation below:

L = mv

where

L = linear momentum,

m = mass, and

v = instantaneous velocity

Momentum is constant if the net external force is zero. This can be expressed mathematically as:

Momentum is a vector quantity. If the components of the momentum are known, the components can be added together (using vector addition) to determine the resultant momentum. Conservation of momentum applies to the components of momentum and can be represented by equations for the 3 dimensions (vertical, horizontal – forward & backward, and horizontal – side to side)

According to the conservation of momentum principle, the total momentum of a system of objects is constant if the net external force acting on the system is zero. This principle is represented mathematically in the equation below:

The conservation of momentum principle is especially useful for analyzing collisions. Collisions are common in sport: Baseballs collide with bats, tennis balls hit rackets. The outcome of these collisions can be explained with the conservation of momentum principle.

• Elastic Collisions

When 2 objects collide in a head-on collision, their combined momentum is conserved.

Example – 2 marbles of the same weight

Imagine 2 marbles are placed 5cm apart on a hard surface. One marble (Marble A) is flicked into the other marble (Marble B) so that it strikes Marble B directly on the centre and not to one side or the other. Just after the collision, Marble A stops or barely moves, and Marble B now moves in the same direction and with about the same velocity as Marble A before the collision. Momentum is conserved, when Marble A strikes Marble B, it transfers its momentum to Marble B.

After the collision, we observed that velocity of Marble A was zero. The initial velocity of marble B was zero, and the final velocity of marble A was zero, so:

The masses of the marbles are equal, so:

For head-on, perfectly elastic collision of 2 objects with equal mass, the precollision momentum of each object is totally transferred to the other project post collision:

• Inelastic Collisions

Not all collisions are perfecting elastic. The opposite of a perfectly elastic collision is a perfectly inelastic collision. In a perfectly inelastic collision, momentum is still conserved, but rather than bouncing off each other, the objects in the collision stay together after the collision and move together with the same velocity. The equations that follow describe the motion of two objects involved in a linear, perfectly inelastic collision.

Example: American Football collisions

Suppose an 80 kg fullback collides in midair with a 120 kg linebacker at the goal line during a goal line stand. Just before the collision, the fullback had a velocity of6 m/s toward the goal line, and the linebacker had a velocity of 5 m/s in the opposite direction. If the collision was perfectly inelastic, would the fullback be moving forward

and score a touchdown just after the collision, or would the defense prevail?

The fullback won’t score. He and the linebacker will be moving back away from the goal line with a velocity of 0.6 m/s.

Most collisions in sports are neither perfectly elastic nor perfectly inelastic but are somewhere in between. These are elastic collisions, but not perfectly elastic

collisions. The coefficient of restitution is a means of quantifying how elastic the collisions of an object are.

Newton’s 2^nd Law of Motion

Newton’s second law of motion states that the change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed. Simply stated, Newton’s second law says that if a net external force is exerted on an object, the object will accelerate in the direction of the net external force, and its acceleration will be directly proportional to the net external force and inversely propotional to its mass. This can be stated mathematically as:

Equation 3.2 can be represented by equations for the three dimensions (vertical, horizontal—forward and backward, and horizontal—side to side).

Newton’s second law expresses a cause-and-effect relationship. Forces cause acceleration. Acceleration is the effect of forces. If a net external force acts on an object, the object accelerates. If an object accelerates, a net external force must be acting to cause the acceleration. Newton’s first law of motion is really just a special case of Newton’s second law of motion—when the net force acting on an object is zero, its acceleration is also zero. Any time an object starts, stops, speeds up, slows down, or changes direction, it is accelerating, and net external force is acting to cause this acceleration.

Impulse and Momentum

In sports and human movement, we are often more concerned with the final outcome resulting from external forces acting on an athlete or object over some duration of time than with the instantaneous acceleration of the athlete or object at some instant during the force application. We want to know how fast the ball was going after the pitcher exerted forces on it during the pitching actions. Newton’s second law can be used to determine this. We can consider what average acceleration is caused by an average net force:

This is the impulse–momentum relationship. Impulse is the product of force and the time during which the force acts. If the force is not constant, impulse is the average force times the duration of the average force. The impulse produced by a net force acting over some duration of time causes a change in momentum of the object upon which the net force acts. To change the momentum of an object, either its mass or its velocity must change. In sports and human movement, most objects we deal with have a constant mass. A change in momentum thus implies a change in velocity. When Newton stated his second law of motion, he really meant momentum when he said motion. The change in momentum of an object is proportional to the force impressed.

The average net force acting over some interval of time will cause a change in momentum of an object. We can interpret change in momentum to mean change in velocity, because most objects have constant mass. If we want to change the velocity of an object, we can produce a larger change in velocity by having a larger average net force act on the object or by increasing the time during which the net force acts.

• Using Impulse to Increase Momentum

The task in many sport skills is to cause a large change in the velocity of something. In throwing events, the ball (or shot, discus, javelin, or Frisbee) has no velocity at the beginning of the throw, and the task is to give it a fast velocity by the end of the throw. We want to increase its momentum. The technique used may be explained in part by the impulse-momentum relationship. A large change in velocity is produced by a large average net force acting over a long time interval.

Example: Throwing a ball with only your wrist vs. with no constraints

You exert the largest impulse on the ball when you used your normal throwing technique. As a result, the ball’s momentum changes very much and ball left your hand with the fastest velocity. The large impulse was the result of a relatively large

average force being exerted on the ball for a relatively long time. The normal throwing technique involved more limbs in the throwing action, and you were able to increase the time during which you could exert a force on the ball (and you were probably able to exert a larger average force). The end result was a faster throw. Due to a longer period of force application, the ball had more time to speed up, and thus its velocity at release was faster.

An important thing to remember about the impulse–momentum relationship

is that the average net force in the impulse term is a vector, as are the velocities, in momentum term. An impulse will cause a change in momentum, and thus a change in velocity, in the direction of the force. If you want to change the velocity of an object in a specific direction, the force you apply or some component of that force, must be in that specific direction.

In many sport situations, the goal is to impart a fast velocity to an object. The initial velocity of the object is zero, and the final velocity is fast, so we want to increase its momentum. We accomplish this by exerting a large force against the object for as long a time as possible (by exerting a large impulse). Techniques in sport activities such as throwing or jumping are largely based on increasing the time of force application to obtain a large impulse.

• Using Impulse to Decrease Momentum

In certain other activities, an object may have a fast initial velocity and we want to decrease this velocity to a slow or zero final velocity. We want to decrease its momentum.

Example – Landing from a jump (flexed knees, hips, and ankles vs stiff legged)

The right side of this equation is the same whether you jump from the chair and land stiff-legged or flex your legs. Your mass, m, does not change. Your final velocity, vf, is the same for both conditions. This is your velocity at the end of the landing, which will be zero. Your initial velocity, vi, is also the same for both conditions if you jump from the same height. This is your velocity when your feet first make contact with the ground. Your change in velocity, (vf − vi), is the same for both conditions. So your change in momentum, m(vf − vi), the right side of equation is the same whether you land stiff-legged or flex your legs.

As long as the change in momentum is the same, changes in impact time are accompanied by inversely proportional changes in average impact force. Thus landing with flexed legs will increase the impact time as compared to stiff-legged landing, leading to lesser impact force as compared to stiff-legged landing.

Newton’s 3^rd Law of Motion

The law states that “To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal and directed to contrary parts”. More simply stated, the Newton’s 3^rd law of motion says that if an object (A) exerts a force on another object (B), the other object (B) exerts the same force on the first object (A) but in the opposite direction. So forces exist in mirrored pairs. The effects of these forces are not cancelled by each other because they act on different objects. Another important point is that it is the forces that are equal but opposite, and not the effects of the forces.

Newton’s Law of Gravitation

Newton’s law of universal gravitation gives us a better explanation of weight. He presented this law in two parts. First, he stated that all objects attract each other with a gravitational force that is inversely proportional to the square of the distance between the objects. Second, he stated that this force of gravity was proportional to the mass of each of the two bodies being attracted to each other. The universal law of gravitation can be represented mathematically as:

The earth’s gravitational force acting on an object is the object’s weight. For an object close to the earth’s surface, several of the terms in equation 3.30 are constant. These terms are G, the universal constant of gravitation; m₂, the mass of the earth; and r, the distance from the centre of the earth to its surface. If we introduce a new constant,

Summary

The basis of linear kinetics, explaining the causes of linear motion, lie in Newton’s laws of motion.

• Newton’s first law explains that objects do not move or do not change their motion unless a net external force act on them.
• An extension of the first law is the conservation of momentum principle.
• Newton’s second law explains what happens if a net external force does act on an object. It will accelerate in the direction of the net external force and acceleration will be inversely related to its mass.
• The impulse-momentum relationship presents Newton’s second law in a way that is more applicable to sports and human movement. Increasing the duration of force application increases the change in momentum.
• Newton’s third law explains that forces acts in pairs. For every force acting on an object, there is another equal force acting in the opposite direction on another object.
• Newton’s law of universal gravitation gives us the basis for the force of gravity.