Motion Notes

Motion

Introduction

• In everyday life, some objects are at rest while others are in motion.
• Motion is perceived when an object's position changes with time.
• Sometimes motion is inferred through indirect evidence, like the movement of air observed through dust or leaves.
• The phenomena of sunrise, sunset, and changing seasons are due to the Earth's motion, which we don't directly perceive.
• An object's motion is relative; it may appear to be moving to one person and stationary to another.
• Most motions are complex, involving straight lines, circular paths, rotation, or vibration, or a combination of these.
• The chapter focuses on describing motion along a straight line using equations and graphs, and later discusses circular motion.

Describing Motion

• The location of an object is described by specifying a reference point called the origin.
• For example, a school is 2 km north of the railway station, where the railway station is the reference point.

Motion Along a Straight Line

• The simplest type of motion is motion along a straight line.
• Consider an object moving along a straight path, starting from point O (the reference point).
• Points A, B, and C represent the object's position at different times.
• The object moves from O to A, then back to C through B.
• Distance: The total path length covered by the object.
  • In the example, the distance covered is OA + AC = $60 km + 35 km = 95 km$.
  • Distance is described by specifying only the numerical value (magnitude).
• Displacement: The shortest distance measured from the initial to the final position of an object.
  • It has both magnitude and direction.
  • In the example, the displacement from O to C is the shortest distance between O and C.
• The magnitude of displacement can be equal to the distance traveled by an object when the object moves along a straight line without changing direction.
  • For motion from O to A, the distance and magnitude of displacement are both $60 km$.
• The magnitude of displacement is not equal to the path length when the object changes direction during its motion.
  • For motion from O to A and back to B, the distance covered is $60 km + 25 km = 85 km$, while the magnitude of displacement is $35 km$.
• The magnitude of displacement for a course of motion may be zero, but the corresponding distance covered is not zero.
  • If the object travels back to O, the displacement is zero but the distance covered is $OA + AO = 60 km + 60 km = 120 km$.
• Distance and displacement are different physical quantities.
• Automobiles use an odometer to show the distance traveled.

Uniform Motion and Non-Uniform Motion

• Uniform Motion: An object covers equal distances in equal intervals of time.
  • Example: An object travels $5 m$ in each second.
• Non-Uniform Motion: An object covers unequal distances in equal intervals of time.
  • Example: A car moving on a crowded street or a person jogging in a park.

Measuring the Rate of Motion

• Different objects may take different amounts of time to cover a given distance.
• The rate at which objects move can be different.
• One way to measure the rate of motion is to find the distance traveled in unit time, which is called speed.
• The SI unit of speed is metre per second (m/s).
• Other units of speed include centimetre per second (cm/s) and kilometre per hour (km/h).
• The speed of an object need not be constant; in most cases, objects are in non-uniform motion.
• Average Speed: The total distance traveled divided by the total time taken.
  • $average \ speed = \frac{Total \ distance \ travelled}{Total \ time \ taken}$
  • If an object travels a distance $s$ in time $t$, then its speed $v$ is: $v = \frac{s}{t}$
  • Example: A car travels $100 km$ in $2 h$. Its average speed is $50 km/h$.

Speed with Direction

• If we specify the direction of motion along with speed, the quantity is called velocity.
• Velocity is the speed of an object moving in a definite direction.
• The velocity of an object can be uniform or variable.
• Velocity can be changed by changing the object's speed, direction of motion, or both.
• Average Velocity: When an object is moving along a straight line at a variable speed, we express the magnitude of its rate of motion in terms of average velocity.
  • Calculated the same way as average speed if the direction does not change.
  • If the velocity of the object is changing at a uniform rate, then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time.
    • $average \ velocity = \frac{initial \ velocity + final \ velocity}{2}$
    • Mathematically, $v<em>{av} = \frac{u + v}{2}$, where $v</em>{av}$ is the average velocity, $u$ is the initial velocity, and $v$ is the final velocity of the object.
• Speed and velocity have the same units (m/s).

Rate of Change of Velocity

• During uniform motion along a straight line, the velocity remains constant with time.
• In non-uniform motion, velocity varies with time.
• Acceleration: A measure of the change in the velocity of an object per unit time.
  • $acceleration = \frac{change \ in \ velocity}{time \ taken}$
  • If the velocity of an object changes from an initial value $u$ to the final value $v$ in time $t$, the acceleration $a$ is: $a = \frac{v - u}{t}$
  • This kind of motion is known as accelerated motion.
  • Acceleration is positive if it is in the direction of velocity and negative when it is opposite to the direction of velocity (deceleration or retardation).
  • The SI unit of acceleration is $m/s^2$.
• Uniform Acceleration: If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time.
  • Example: The motion of a freely falling body.
• Non-Uniform Acceleration: If an object's velocity changes at a non-uniform rate.
  • Example: A car traveling along a straight road increases its speed by unequal amounts in equal intervals of time.

Graphical Representation of Motion

• Graphs provide a convenient method to present basic information about motion.

Distance-Time Graphs

• The change in the position of an object with time can be represented on a distance-time graph.
  • Time is taken along the x-axis, and distance is taken along the y-axis.
• Uniform Speed: When an object travels equal distances in equal intervals of time, the distance traveled is directly proportional to time taken.
  • A graph of distance traveled against time is a straight line.
• The distance-time graph can be used to determine the speed of an object.
  • $v = \frac{s<em>2 - s</em>1}{t<em>2 - t</em>1}$
• Non-Uniform Speed: The distance-time graph is a curve.

Velocity-Time Graphs

• The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph.
  • Time is represented along the x-axis, and velocity is represented along the y-axis.
• Uniform Velocity: The height of its velocity-time graph will not change with time and is a straight line parallel to the x-axis.
  • The area enclosed by the velocity-time graph and the time axis will be equal to the magnitude of the displacement.
• Uniformly Accelerated Motion: The velocity-time graph is a straight line.
  • The area under the velocity-time graph gives the distance (magnitude of displacement) moved by the car in a given interval of time.
• Non-Uniformly Accelerated Motion: Velocity-time graphs can have any shape.

Equations of Motion

• When an object moves along a straight line with uniform acceleration, its velocity, acceleration, and the distance covered can be related by a set of equations known as the equations of motion.
• $v = u + at$
• $s = ut + \frac{1}{2}at^2$
• $2as = v^2 - u^2$
  • Where:
    • $u$ is the initial velocity.
    • $v$ is the final velocity.
    • $a$ is the uniform acceleration.
    • $t$ is the time.
    • $s$ is the distance traveled.

Uniform Circular Motion

• When the velocity of an object changes, the object is accelerating.
• The change in velocity could be due to change in its magnitude or the direction of the motion or both.
• Uniform Circular Motion: When an object moves in a circular path with uniform speed.
  • The only change in velocity is due to the change in the direction of motion.
  • The motion is an example of accelerated motion.
  • The circumference of a circle of radius $r$ is given by $2\pi r$.
  • If the athlete takes $t$ seconds to go once around the circular path of radius $r$, the speed $v$ is given by:
    • $v = \frac{2 \pi r}{t}$
• Examples: The motion of the moon and the earth, a satellite in a circular orbit around the earth, a cyclist on a circular track at constant speed.