Table Of ContentMust know ForMula on Physics' For aiPMt
Parallelogram law of vector addition :
R =
Let α be the angle, which the resultant makes with
Resolution of vectors
Scalar Product (Dot Product) :
If and are two vectors having angle θ between them, then their scalar (Dot) product written
as and read as dot is defined as
Properties of Scalar Product
(i) It is always a scalar and it will be positive if angle between them is acute, negative if angle
between them is obtuse and zero if angle between them is 90°.
(ii) It obeys commutative law
(iii) It obey distributive law
(iv) By definition
cos θ = ⇒ θ =
where θ is the angle between two vectors.
(v) Scalar (Dot) product of two mutually perpendicular vectors is zero i.e.,
(2)
Must know ForMula on Physics' For aiPMt
(vi) Scalar (Dot) product will be maximum when θ = 0° i.e.,
vectors are parallel to each other.
(vii) If and are unit vectors, then and
(viii) Dot product of unit vectors
(ix) Square of a vector
(x) If the two vectors and , in terms of their rectangular components, are
and , then
= A B + A B + AB
x x y y z z
Vector Product (Cross Product)
If and are two vectors, then their vector product is written as is vector and is read as
cross . It is defined as
......(7)
where is unit vector along the direction of
Properties of Vector Product
(i) Cross product of two vectors is not commutative
;
(ii) Cross product is not associative
(iii) Cross product obey distributive law
(iv) If θ = 0 or π it means two vectors are collinear.
and conversely, if , then the vector are parallel provided are non-zero
vectors.
(3)
Must know ForMula on Physics' For aiPMt
This may be regarded as a test to decide whether the given two vectors are parallel or not.
(v) If θ = 90°, then
(vi) The vector product of any vector with itself is
(vii) If , then or or
(viii) If and are unit vectors, then
(ix) Cross product of unit vectors
(4)
Must know ForMula on Physics' For aiPMt
(5)
Must know ForMula on Physics' For aiPMt
Representation of errors
Errors can be expressed in the following ways :
Absolute
Propagation of errors in mathematical operations
Rule I :
If X = A + B or X = A – B and if ±∆A and ±∆B represent the absolute errors in A and B respectively,
then maximum absolute error in X = ∆X = ∆A + ∆B and
Maximum percentage error = ....(6)
The result will be written as X ± ∆X (in terms of absolute error)
or X %(in term of percentage error)
Rule II :
If X = A × B or X = A/B
then ....(7)
Rule III :
If X = An then
If X = ApBqCr then
(6)
Must know ForMula on Physics' For aiPMt
Motion in Straight Line
Speed and Velocity
Speed is related to distance and it is a scalar while velocity is related to displacement and it is
a vector.
For a moving body speed can’t have zero or negative values but velocity can have.
Average Velocity
Let a particle is at a point A at time t and B at time t . Position vectors of A and B are and .
1 2
The displacement in this time interval is the vector = . The average velocity in this time
interval is,
here = change in position vector.
The direction of average velocity is in the direction of displacement .
Average Speed
Average speed is defined for a time interval and is given by the total distance traversed in the given
time interval divided by the time interval.
Average speed of a trip v =
av
If a particle travels a distance S in time t to t , the average speed is v =
1 2 av
Acceleration
The motion of an object is said to be accelerated if its velocity changes with time (when either
the speed changes or the direction of motion changes). The acceleration is defined as the
rate of change of velocity vector w.r.t. time.
Average Acceleration
During an interval of time when the velocity of particle changes from to , the average
acceleration is defined as the change in velocity divided by interval of time (say ∆t).
Average acceleration = ⇒
The magnitude of average acceleration
The Acceleration-time Graph
Acceleration time curves give information about the variation of acceleration with time. Area under
the acceleration time curve gives the change in velocity of the particle in the given time interval.
(7)
Must know ForMula on Physics' For aiPMt
Analysis of Uniformaly Accelerated Motion
Case I : For uniformaly accelerated motion with initial velocity u and initial position x .
0
Velocity time graph
In every case, tan θ = a
0
Position time graph
Initial position x of the body in every case is x (> 0)
0
Case II : For uniformly retarded motion with initial velocity u and initial position x .
0
Velocity time graph
In every case tan θ = – a
0
Position time graph
Initial position x of the body in every case is x (>0)
0
Concepts
1. Slopes of v-t or s-t graphs can never be infinite at any point, because infinite slope of v-t graph
means infinite acceleration. Similarly, infinite slope of s-t graph means infinite velocity. Hence,
the following graphs are not possible.
2. At one time, two values of velocity or displacement are not possible Hence, the following
graphs are not acceptable.
(8)
Must know ForMula on Physics' For aiPMt
3. The displacement-time graph cannot take sharp turns because it gives two different velocities
at that point.e.g.
4. The displacement-time graph cannot be symmetric about the time-axis because at an in-
stant a particle cannot have two displacements. But the graph may be symmetric about the
displacement-axis. e.g.
5. The distance-time graph is always an increasing curve for a moving body.
6. The displacement-time graph does not show the trajectory of the particle.
If a body dropped from some height (initial velocity zero)
(i) Equation of motion : Taking initial position as origin and direction of motion
(i.e., downward direction) as a positive, here we have
u = 0 [As body starts from rest]
a = +g [As acceleration is in the direction of motion
v = g t …(i)
…(ii)
v2 = 2gh …(iii)
...(iv)
(ii) Graph of distance, velocity and acceleration with respect to time :
(iii) As h = (1/2)gt2, i.e., h ∝ t2, distance covered in time t, 2t, 3t, etc., will be in the ratio of 12 :
22 : 32, i.e., square of integers.
(iv) The distance covered in the nth sec,
So distance covered in I, II, III sec, etc., will be in the ratio of 1 : 3 : 5, i.e., odd integers only.
If a body is projected vertically downward with some initial velocity
Equation of motion : v = u + g t
v2 = u2 + 2 gh
(9)
Must know ForMula on Physics' For aiPMt
If a body is projected vertically upward
(i) Equation of motion : Taking initial position as origin and direction of motion (i.e., vertically
up) as positive
a = – g [As acceleration is downwards while motion upwards]
So, if the body is projected with velocity u and after time t it reaches up to height h then
v = u – g t ; ; v2 = u2 – 2gh ;
(ii) For maximum height v = 0
So from above equation
u = gt,
and u2 = 2gh
(iii) Graph of distance, displacement, velocity and acceleration with respect to time (for
maximum height) :
It is clear that both quantities do not depend upon the mass of the body or we can say that in
absence of air resistance, all bodies fall on the surface of the earth with the same rate.
Applications of Relative Velocity
Relative velocity concept is very useful in
(1) Motion analysis between two bodies. (2) Rain / wind / aircraft based problems.
(3) River (Swimmer ) based problem.
Swimming into the river
A man can swim with velocity , i.e., it is the velocity of man w.r.t. still water.
If water is also flowing with velocity then velocity of man relative to ground
(i) If the swimming is in the direction of flow of water or along the downstream then
v = v + v
m R
(ii) If the swimming is in the direction opposite to the flow of water or along the upstream then
(10)
Must know ForMula on Physics' For aiPMt
v = v – v
m R
(iii) If man is crossing the river i.e., and not collinear then use the vector algebra
For shortest path :
If man wants to cross the river such that his “displacement should be minimum”. It means he wants
to reach just opposite point across the river. Man should start swimming at an angle θ with the
perpendicular to the flow of river towards upstream.
So that its resultant velocity . It is in the direction of displacement AB.
To reach at B v sin θ = v ⇒ sin θ =
R
component of velocity of v along AB is = vcos θ. so time taken
For minimum time
To cross the river in minimum time the velocity along AB (v cos θ) should be maximum. It is possible
if θ = 0, i.e., swimming should start perpendicular to water current. Due to effect of river velocity,
man will reach at point C along resultant velocity, i.e., his displacement will not be minimum but
time taken to cross the river will be minimum.
In time t swimmer travels distance BC along the river with speed of river v ∴ BC = t v Distance
min R min R
travelled along river flow = drift of man = t v =
min R
Oblique Projectile Motion
Equation of Trajectory : A projectile thrown with velocity u at an angle θ with the horizontal. The
velocity u can be resolved into two rectangular components.
(11)
Description:MUST KNOW FORMULA ON PHYSICS' FOR AIPMT. Parallelogram law of vector
addition : R = Let α be the angle, which the resultant makes with. Resolution of
