Motion In A Plane-FORMULA SHEETS | PHYSICS

Formula Sheet

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Motion In A Plane

- All the formulas in one go

Basic of Vector

Scalars

Those physical quantities which require only magnitude but no direction for their complete representation, are called scalars.

Distance, speed, work, mass, density, etc are the examples of scalars.

Vectors

Those physical quantities which require magnitude as well as direction for their complete representation and follows vector laws are called vectors.

Tensors

Tensors are those physical quantities which have different values in different directions at the same point.

Moment of inertia, radius of gyration, modulus of elasticity, etc are the examples of tensors.

Vector Laws 1.Addition of Vectors

Triangle Law of Vectors

If two vectors

A

and

B

acting at a point are inclined at an angle

θ

, then their resultant

R = A^{2} + B^{2} + 2 A B c o s θ

and if the resultant vector

R

subtends an angle

α

with vector

A

, then

t a n α = \frac{B s i n θ}{A + B c o s θ}

Parallelogram Law of Vectors

Resultant of vectors

A

and

B

is given by

R = A^{2} + B^{2} + 2 A B c o s θ

and if the resultant vector

R

subtends an angle

β

with vector

A

, then

t a n β = \frac{B s i n θ}{A + B c o s θ}

Polygon Law of Vectors

R = A + B + C + D

2. Subtraction of Vectors

3. Multiplication of a Vector

When a vector A is multiplied by a real number n, then its magnitude becomes n times but direction and unit remains unchanged.

R = n A

and

∣ R ∣ = n ∣ A ∣

Scalar or Dot Product of Two Vectors

The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them. It is denoted by . (dot).

A . B = ∣ A ∣ ∣ B ∣ c o s θ

Vector or Cross Product of Two Vectors

The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by * (cross).

A \times B = ∣ A ∣ ∣ B ∣ s i n θ \overset{n}{^}

The direction of unit vector

n

can be obtained from right hand thumb rule.

Motion in a Plane

Position

R = x \hat{i} + y \hat{j}

Velocity

V = V_{x} \hat{i} + V_{y} \hat{j}

∣ V ∣ = (V_{x})^{2} + (V_{y})^{2}

Acceleration

a_{x} = \frac{d ( V _{x} )}{d t}

a_{y} = \frac{d ( V _{y} )}{d t}

Motion in Plane Equations

For $X - a x i s$

v_{x} = u_{x} + a_{x} t

s_{x} = u_{x} t + \frac{1}{2} a_{x} t^{2}

v_{x}^{2} = u_{x}^{2} + 2 a_{x} s_{x}

Here,

u_{x}

= initial velocity along the

X - a x i s

v_{x}

= final velocity along the

X - a x i s

a_{x}

= acceleration along the

X - a x i s

s_{x}

= displacement along the

X - a x i s

t

= time taken while executing the motion along the

X - a x i s

For $Y - a x i s$

v_{y} = u_{y} + a_{y} t

s_{y} = u_{y} t + \frac{1}{2} a_{y} t^{2}

v_{y}^{2} = u_{y}^{2} + 2 a_{y} s_{y}

Here,

u_{y}

= initial velocity along the

Y - a x i s

v_{y}

= final velocity along the

Y - a x i s

a_{y}

= acceleration along the

Y - a x i s

s_{y}

= displacement along the

Y - a x i s

t

= time taken while executing the motion along the

Y - a x i s

Relative Velocity in Two Dimensions

Suppose that two objects

A

and

B

are moving with velocities

v_{A}

and

v_{B}

(each with respect to some common frame of reference, say ground.). Then, velocity of object

A

relative to that of

B

is :

v_{A B} = v_{A} - v_{B}

CROSSING RIVER

A boat or man in a river always moves in the direction of resultant velocity of velocity of boat (or man) and velocity of river flow.

1. Shortest Time :

Velocity along the river,

v_{x} = v_{R}

Velocity perpendicular to the river,

v_{f} = v_{m R}

The net speed is given by

v_{m} = v_{m R}^{2} + v_{R}^{2}

2. Shortest Path :

Velocity along the river,

v_{x} = 0

Velocity perpendicular to river

v_{y} = v_{m R}^{2} - v_{R}^{2}

The net speed is given by

v_{m} = v_{m R}^{2} - v_{R}^{2}

At an angle of

9 0^{o}

with the river direction, velocity

v_{y}

is used only to cross the river,

therefore time to cross the river,

t = \frac{d}{v _{y}} = \frac{d}{v _{m R}^{2} - v _{R}^{2}}

and velocity

v_{x}

is zero, therefore, in this case the drift should be zero.

\Rightarrow

v_{R} - v_{m R} s i n θ = 0

v_{R} = v_{m R} s i n θ

θ = s i n^{- 1} (\frac{v _{R}}{v _{m R}})

Rain Problems

V_{R m} = V_{R} - V_{m}

V_{R m} = V^{2} - R + V_{m}^{2}

Projectile Motion

Time of flight :

T = \frac{2 u s i n θ}{g}

Horizontal range :

R = \frac{u ^{2} s i n 2 θ}{g}

Maximum height :

H = \frac{u ^{2} s i n ^{2} θ}{2 g}

Trajectory equation (equation of path) :

y = x t a n θ - \frac{g x ^{2}}{2 u ^{2} c o s ^{2} θ} = x t a n θ (1 - \frac{x}{R})

Projection on an inclined plane

Uniform Circular Motion

The angular speed

ω

, is the rate of change of angular distance. It is related to velocity

v

v = ω R

Centripetal acceleration =

a_{c} = ω^{2} R = \frac{v ^{2}}{R}

T

is the time period of revolution of the object in circular motion and

f

is its frequency, we have

ω = 2 π f

v = 2 π f R

a_{c} = 4 π^{2} f^{2} R

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