Physics: Rectilinear Motion & Newton's Laws

Rectilinear Motion

• Rectilinear motion covers pages 6 to 46
• Newton's Laws of Motion spans pages 47 to 78

Fundamental Definitions and Concepts

• Motion involves the change in body position over a period of time
• Motion and rest are relative
• A rider can be still relative to other riders but in motion relative to those outside the train
• Displacing motion is when the body moves from a start to an end point
  • Can be straight like an object falling from a window
  • Or it can be a curving line, like projectiles
• Circular and oscillator motions are outside the scope of these notes
• A particle is a virtual point which describes the motion of a body while ignoring internal or circular motion

Position, Displacement, and Distance

• The position vector of a particle is a vector coinciding with the observer's position at the body's location
• The position vectorr = Xi + Yj
• Consider a car that moves from an initial point A to a final point B along a path
  • Displacement refers to the vector AB
  • Starting point A coincides with the initial position
  • End point B coincides with the end position
• The magnitude of displacement is the distance between initial and final positions:|AB| = |S|
• The direction of displacement follows the motion from start to end
• Covered distance is the length of the path taken, and it is a scalar quantity
• The magnitude of displacement equals the covered distance when motion occurs along a straight line in a single direction
• If a body returns to its starting point, the magnitude of displacement is zero
• The magnitude of displacement ≤ the covered distance.

Example 1: Displacement Problem

• A body moves 12 meters East, followed by 5 meters North, then stops.
• The covered distance is 12 + 5 = 17 meters
• Displacement is represented by directed line segment OB
  • Its magnitude = (5^2 + 12^2)^0.5 = 13 meters
  • Direction = tan-1(5/12) = 22° 37' 12" North of East
• The norm of the displacement = 13 meters
  • Direction is 22° 37' 12" (North of East)

The Relation Between the Position Vector and Displacement

• Let O be the observer's position and A and B are the body's start and end points
• Let r0 be the starting position at time t and r be the ending position at time (t + h).
• The displacement S = r - r0 = (X2 - X1)i + (Y2 - Y1)j
• The norm of S is ||S|| = √((X2 - X1)^2 + (Y2 - Y1)^2)
• For a unit vector ê in the direction of AB, then S = ||S|| ê

Example 2 : Position Vector Problem

• A body moves such that its position vector r = (t + 2)i + (3t - 2)j
• S = r - r0 = ti + 3tj
• The norm of the displacement until t = 4 seconds is √(4^2 + (3*4)^2) = √(16 + 144) = √160
• At t=2, r2 = 4i+4J ; at t=4, r4 = 6i+10j
• The displacement between t = 2 and t = 4 is r4+r2 = 2i + 6j, length unit
• The norm = √(2^2 + 6^2) = √(4+36) = √40 length unit

Velocity and Speed

• Velocity expresses the rate of change in a body's position with respect to time but is a vector
• Speed is a scalar quantity which expresses the magnitude/norm of the velocity
• Velocity is equal to the magnitude/speed, plus the direction of motion
• 90 km/hr expresses speed
• Expressing 90 km/hr in the direction of north expresses the velocity

Speed Units

• km / hr, metre / second (m/sec), and centimetre / second (cm/sec)
• 1 km/hr = 5/18 m/sec
• 1 km/hr = 250/9 cm/sec
• From km/hr to m/sec multiply by 5/18
• From km/hr to cm/sec, multiply by 250/9

The Uniform Motion

• The state in which both magnitude and direction of the velocity are constant
• As the particle moves along a straight line
• Means the body covers equal distances for same period
• The straight motion occurs in a straight line
• The norm of the displacement equals the covered distance s = tv
• Requires constant velocity, implying the object covers equal displacements over equal time intervals
• Constant speed is denoted by |v|

Variable Motion

• In variable motion, each of the direction or the magnitude of the velocity or both could vary over time
• A car moving at 80 km/hr along a circular path has constant speed but the direction changes
• In straight line motion, assume a unit vector ê in the direction to where the object is going
  • s = ||S|| if displacement is in the same direction, and -|S|| if it's in the opposite direction
  • v = ||v|| if the velocity's direction is the same, and -||v|| if it's opposite.

Average Speed and Average Velocity

• The average speed within a quotient of total distance in interval over time interval
• Average velocity is a vector quantity
• The average velocity of VA = (R2 - R1)/(t2 - t1)
• It is not necessary that the average speed is equal to the norm of the average velocity unless the body is moving in the same direction
• The physical concept of the average speed It is the speed that if the body would mo