Linear and Angular Motion
Kinematics
Kinematics is the branch of mechanics that studies motion without considering the forces that cause it. It focuses on describing motion using parameters such as position, displacement, speed, velocity, and acceleration.
Motion
Motion is the change in position of an object or body over time. It can involve movement from one location to another or changes in the position of individual body parts relative to each other. Motion is described using key concepts such as displacement, velocity, and acceleration.
Motion can be:
- Linear (motion in a straight line), such as a sprinter running down a track.
- Curvilinear (motion along a curved path), such as a football being kicked in a long arc.
- Angular or rotational (motion around an axis), such as a figure skater spinning on one foot.
- General (a combination of linear and angular motion), such as a cyclist pedaling while moving forward.
Linear Motion
Linear motion refers to movement along a straight path.
It can be described using three key terms:
- Speed: The distance covered per unit of time. It's a scalar quantity, meaning it has no direction.
- Velocity: The rate of change of displacement (distance in a specific direction) over time. It's a vector quantity, meaning it includes both magnitude and direction.
- Acceleration: The rate of change of velocity over time. It can be positive (speeding up) or negative (slowing down).
- A sprinter running 100 meters in 10 seconds has a speed of 10 m/s.
- If they start from rest and reach this speed in 2 seconds, their acceleration is 5 m/s².
Measurements and Position
Position
Position refers to the location of an object or body, typically given by its coordinates.
Coordinates
Coordinates measure distance from an origin (e.g., in meters) and are given in two dimensions (x, y: horizontal and vertical) or three dimensions (x, y, z: horizontal, vertical, lateral). Two systems for three-dimensional coordinates: System 1: x = horizontal, y = vertical, z = lateral and System 2: x = horizontal, y = lateral, z = vertical. Angular coordinates are measured as angles around one or more axes.
Newton's laws of motion and linear motion
1. Newton's first law: the law of inertia
Newton's First Law
An object will remain at rest or keep moving at a constant velocity unless acted upon by an outside force. This law is also known as the law of inertia
- This law emphasizes that a force is necessary to change the motion of an object.
- This law explains why it is harder to start or stop an object's motion.
- The resistance to change in motion is called inertia.
Inertia
Inertia is a property of matter that causes an object to resist changes in its state of motion. The more mass an object has, the greater its inertia.
- A soccer ball will not move unless kicked (object at rest) and will continue to roll in a straight line unless acted upon by external forces like friction or a player kicking it again (object in motion).
- A hockey puck glides across the ice until friction or a player's stick changes its motion.
Application in Sports
- Inertia explains the challenge faced by athletes when starting and stopping motion.
- It takes significant force to overcome inertia and accelerate a stationary object or decelerate a moving object.
- Running: A sprinter at the starting line has to apply force to overcome inertia and accelerate from rest. The sprinter’s body tends to stay at rest until sufficient force is applied.
- Football: A football, once kicked, continues in motion until external forces (e.g., friction with the ground, air resistance) slow it down or stop it.
- Stopping: It takes more effort to stop a larger object (like a heavy sled) compared to a smaller one because of inertia.
This law explains why athletes need to exert more force to overcome inertia when starting or stopping movement.
2. Newton's second law: the law of acceleration
Newton's Second Law
The force on an object is equal to its mass times its acceleration. This law states that the greater the force applied to an object, the greater the acceleration.
The acceleration of an object is proportional to the force applied and inversely proportional to its mass.
This is expressed as: F = ma
Where:
- F is the force acting on the object (measured in Newtons, N),
- m is the mass of the object (measured in kilograms, kg),
- a is the acceleration of the object (measured in meters per second squared, m/s²).
- This law shows how the force applied to an object influences its motion.
- The greater the force applied to an object, the greater its acceleration, provided that the mass remains constant.
- Students often think more force always means more speed.
- Remember - the object's mass matters!
- A heavy shot put needs much more force than a tennis ball for the same acceleration.
- Acceleration is change in velocity demanded by the time taken.
- So Newton's second law of motion could be rewritten as:
$$F = \frac{m(v - u)}{t}$$
Application in Sports
- To maximize acceleration, athletes aim to increase the force they apply and/or reduce their mass.
- For example, a sprinter can work on strengthening their legs (to increase the force they apply to the ground) or reduce body fat to lower their mass.
Sprinting
- A sprinter needs to apply a large force to the ground to generate a high acceleration from the starting blocks.
- The larger the force they exert, the quicker they can accelerate.
Javelin Throw
- The athlete applies a large force to the javelin, which accelerates it through the air.
- The mass of the javelin limits how much acceleration can be achieved for a given force.
- When working through problems involving acceleration, always look at the force and mass variables.
- If mass increases, for the same force, acceleration will decrease.
- If force increases, acceleration increases (as long as mass remains constant).
3. Newton's third law: the law of action-reaction
Newton's Third Law
When two objects interact, they apply forces to each other of equal magnitude and opposite direction. This law is also known as the law of action and reaction
- The Third Law of Motion states that for every action, there is an equal and opposite reaction.
- The action force is the force that is exerted by one object, and the reaction force is the equal and opposite force exerted by the second object.
- This means that when one object applies a force on another object, the second object applies an equal but opposite force on the first object.
- The action and reaction forces are always equal in magnitude and opposite in direction, but they act on different objects.
- Remember that action and reaction forces act on different objects.
- For example, when jumping, the action force is applied to the ground, and the reaction force acts on the athlete's body.
Application in Sports
- While action and reaction forces are equal in size, they do not cancel out because they act on different objects.
- The force an athlete applies to the ground results in an opposite force that propels them in the opposite direction.
- Jumping: When an athlete pushes off the ground (action), the ground pushes back with an equal force, propelling the athlete upward (reaction).
- Swimming: A swimmer pushes water backward with their hands and feet (action), and the water pushes the swimmer forward (reaction).
- Walking: When a person walks, their foot pushes backward against the ground (action), and the ground pushes them forward (reaction).
- Students may incorrectly assume that action and reaction forces cancel each other out.
- These forces do not cancel because they act on different objects.
Key principles of Linear motion
Stability
The ability to maintain equilibrium and resist changes in position, determined by several factors affecting balance.
Four key factors influence stability in sport:
- Height of center of mass (lower = more stable)
- Size of base of support (wider = more stable)
- Position of line of gravity (should fall within base)
- Total mass (more = greater stability)
Wrestlers demonstrate all stability principles by:
- Lowering their center of mass
- Widening their stance
- Keeping their line of gravity centered
- Using their body mass effectively
Don't confuse stability with balance! Stability is resistance to movement, while balance is maintaining controlled position. A sumo wrestler is very stable but might not have great balance for gymnastics.
Distance and Displacement
Distance
Distance
Distance is the total length of the path traveled by an object, regardless of direction.
Distance is a scalar quantity, meaning it only has magnitude (size) and no direction.
Distance=Total Path Traveled
ExampleIf a runner completes two laps around a 400 m track, the total distance covered is 800 m.
Displacement
Displacement
Displacement is the shortest straight-line distance between an object’s starting and ending positions, including direction.
Displacement is a vector quantity, meaning it has both magnitude and direction.
ExampleIf a runner starts at point A, runs 100 m to point B, then returns 100 m back to point A:
- Distance traveled = 200 m
- Displacement = 0 m (since the runner ended at the starting position)
Think of distance as the number of steps you take during a walk, while displacement is how far you are from where you started.
Linear Speed, Velocity, and Acceleration
Speed
Speed
Speed is the rate of change of distance over time.
- It is a scalar quantity, meaning it only has magnitude and no direction.
- It is measured in meters per second (m/s).
$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$
ExampleA runner covering 100 meters in 10 seconds has a speed of 10 m/s.
NoteSpeed does not indicate the direction of motion, only how fast an object is moving.
Velocity
Velocity
Velocity is the rate of change of displacement and includes direction.
- Unlike speed, velocity is a vector quantity (it has both magnitude and direction).
- Velocity is measure in meters per second (m/s)
- Velocity can be positive, negative, or zero depending on direction.
$$ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} $$
ExampleIf a cyclist moves 100 m east in 10 s, the velocity is 10 m/s east.
Acceleration
Acceleration
Acceleration is the rate of change of velocity over time.
- It describes how quickly an object speeds up, slows down, or changes direction.
- The unit is meters per second squared (m/s²)
$$ \text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Taken}} $$
Types of acceleration:
- Positive acceleration: Speeding up (e.g., a runner increasing speed).
- Negative acceleration (deceleration): Slowing down (e.g., a runner stopping after a sprint).
A sprinter increases velocity from 0 to 8 m/s in 2 seconds.
Relationship Between Force, Acceleration, and Mass
Newton’s Second Law states that the force acting on an object is equal to the product of its mass and acceleration:
F=ma
Where:
- F = Force (Newtons, N)
- m = Mass (kg)
- a = Acceleration (m/s²)
- If mass remains constant, increasing force increases acceleration.
- If force remains constant, increasing mass decreases acceleration.
- A heavier rugby player requires more force to accelerate than a lighter player.
- A baseball pitcher applies more force to throw a fastball than a slow pitch.
- Think of pushing a shopping cart.
- If the cart is empty, it is easy to accelerate.
- If the cart is full of groceries, you need to apply more force to move it at the same acceleration.
- Acceleration occurs whenever velocity changes, even if the speed stays the same.
- If an object moves in a circle at constant speed, it is still accelerating due to the change in direction.
Impulse and momentum
Momentum
Momentum is the product of mass and velocity.
p = m*v
where:
- p=momentum (kgm/s)
- m=mass (kg) and
