# Summary

- 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 Î» is also a vector. Whose magnitude is Î» times the magnitude of the vector A and whose direction is the same or opposite depending upon whether Î» 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:

A+B=B+A - 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:

Î›0 = 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=Î»a+Î¼
- Where Î» and Î¼ 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}+A_{y}

- A=A
- Where A
_{x},A_{y }are its components along x- and y - axes. If vector A makes an angle Î¸ with the x axis then A_{x}=A cos Î¸ ,A_{y }=A sin Î¸ and A=,tan Î¸ = - 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
- Î”r=r'-r
- =(x'-x)+(y'-y)
- =Î”x+Î” y

- Î”r=r'-r
- If an object undergoes a displacement Î”r in time Î”t, its average velocity is given by v=. The velocity of an object at time t is the limiting value of the average velocity as Î”t 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 timeÎ”t, then its average acceleration is given by
- The acceleration a at any time t is the limiting value of as Î”t -> 0
- In component for, we have:a=a
_{x}+a_{y}+a_{z} - 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=r
_{0}+v_{0}t+1/2 at^{2}

- R=r
- And its velocity is given by:
- V=v
_{0}+at - Where v
_{0}is the velocity at time t=0 - In component form:
- X=x
_{0}+v_{ox}t+1/2a_{x}t^{2} - Y=y
_{0}+v_{oy}t+1/2 a_{y}t^{2} - V
_{x}=v_{ox}+a_{x}t - V
_{y}=v_{oy}+a_{y}t - 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 Î¸
_{0}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=(v
_{o}cosÎ¸_{0})t - Y=(v
_{o}sinÎ¸_{0})t-(1/2)gt^{2} - v
_{x}=v_{ox}=v_{o}cos - V
_{y}=v_{0}sin Î¸_{0}-gt

- X=(v
- The path of a projectile is parabolic and is given by:
- Y=(tanÎ¸
_{0})x- - The maximum height that a projectile attains is:
- The time taken to reach this height is:

- Y=(tanÎ¸
- 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 a
_{c}=v^{2}/R.The diretin of a_{c}is always towards the centre of the circle, - The angular speed Ï‰,the rate of change of angular distance. It is related to velocity v by v=Ï‰R. The acceleration is a
_{c}=Ï‰2R. - If T is the time period of revolution of the object in circular motion and v is its frequency, we have Ï‰=2Ï€v, v=2Ï€vR, a
_{c}=4Ï€^{2}v^{2}R.