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  • Scalar quantities are quantities with magnitudes only. Examples are distance,speed, mass and temperaure.
  • Vector quantities are quantities with magnitude and direction both. Examples are displacement, velocity and acceleration. They obey special rules of vector algebra.
  • A vector A multiplied by a real number λ is also a vector. Whose magnitude is λ times the magnitude of the vector A and whose direction is the same or opposite depending upon whether λ is positive or negative.
  • Two vectors A and B may be added graphically using head-to-tall method or parallelogram method.
  • Vector addition is commulative:
    It also obeys the associative law:
  • A null or zero vector is a vector with zero magnitude. Since the magnitude is zero, we don't have to specify its direction. It has the properties:
    A+0 = A
    Λ0 = 0
    0A = 0
  • The subtraction of vector B from A is defined as the sum of A and -B: B=A+(-B)
  • A vector A can be resolved into component along two given vectors a and b lying in the same plane:
    • A=λa+μ
    • Where λ and μ are real numbers.
  • A unit vector associated with a vector A has magnitude one and is along the vector A:n=
  • The unit vectors ,, are vectors of unit magnitude and point in the direction of the x-,y-, and z-axes, respectively in a right-handed coordinate system.
  • A vector A can be expressed as
    • A=A x+Ay
  • Where Ax,Ay are its components along x- and y - axes. If vector A makes an angle θ with the x axis then Ax=A cos θ ,Ay =A sin θ and A=,tan θ =
  • Vectors can be convenietly added using analytical method. If sum of two vectors A and B, that lie in xy plane, is R then:
    • where, and
  • The position vector of an object in x-y plane is given by r=x + y and the displacement from position r to position r' is given by
    • Δr=r'-r
      • =(x'-x)+(y'-y)
      • =Δx+Δ y
  • If an object undergoes a displacement Δr in time Δt, its average velocity is given by v=. The velocity of an object at time t is the limiting value of the average velocity as Δt tends to zero:
  • it can be written in unit vector notation as:
  • where
  • When position of an object is plotted on a coordinated system, v is always tangent to the curve representing the path of the object.
  • If the velocity of an object changes from v to v' in timeΔt, then its average acceleration is given by
  • The acceleration a at any time t is the limiting value of as Δt -> 0
  • In component for, we have:a=ax+ay+az
  • Where
  • If an object is moving in a plane with constant acceleration a=|a|=
  • And its position ector at time t=0 is r0, then at any other time t, it will be at a point given by:
    • R=r0+v0t+1/2 at2
  • And its velocity is given by:
  • V=v0+at
  • Where v0 is the velocity at time t=0
  • In component form:
  • X=x0+voxt+1/2axt2
  • Y=y0+voyt+1/2 ayt2
  • Vx=vox+axt
  • Vy=voy+ayt
  • Motion in a plane can be treated as superposition of two separate simultaneous one-dimensional motions along two perpendicular directions.
  • An object that is in flight after being projected is called a projectile. If an object is projeted with inital velocity vo making an angle θ0 with x-axis and if wer assume its inital position to coincide witht the origin of the coordinate system, then the position and velocity of the projectile at time t are given by:
    • X=(vo cosθ0)t
    • Y=(vo sinθ0 )t-(1/2)gt2
    • vx=vox=vocos
    • Vy=v0sin θ0-gt
  • The path of a projectile is parabolic and is given by:
    • Y=(tanθ0)x-
    • The maximum height that a projectile attains is: 
    • The time taken to reach this height is:
  • The horizontal distance travelled by a projectile from its initial position to the position it passes y=0 during its fall is called the range. R of the projectile. It is:
    • R=
  • When an object follows a circular path at constant speed, the motion of the objet is called uniform circular motion. The magnitude of its acceleration is ac=v2/R.The diretin of ac is always towards the centre of the circle,
  • The angular speed ω,the rate of change of angular distance. It is related to velocity v by v=ωR. The acceleration is ac=ω2R.
  • If T is the time period of revolution of the object in circular motion and v is its frequency, we have ω=2πv, v=2πvR, ac=4π2v2R.

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