Motion in a Plane

Position vector

Vector which gives the position of the object with respect to origin.

Displacement vector

The straight line between the two points irrespective of the path followed. Shows the net distance covered.

Two vectors are equal if

they have the same magnitude and direction

Polar vectors

Vectors having linear effect. Reverses sign when the coordinate axis is reversed.

Equal vectors

Vectors that have the same magnitude and the same direction.

Negative vector

a vector that has the same magnitude as another vector but is pointing in the opposite direction

Unit vector

A vector with a magnitude of 1

Zero vector

Vector of length 0, thus has all components equal to zero.

Coplanar vectors

Vectors which are lying on the same in a three-dimensional plane. They are parallel to the same plane.

Co-initial vectors

When two or more vectors have the same starting point

Null vector

Adding two vectors with equal magnitudes and opposite directions results in null vector.
A + (-A) = 0

The angle between two parallel vectors acting in the same direction is

0°

The angle between two parallel vectors acting in opposite directions is

180°

Triangle law of vector addition

If two sides of a triangle are represented by two vectors in the same order, then the resultant is represented both in magnitude and direction by the third side of the triangle taken in opposite order

Parallelogram law of vector addition

When two co-initial vectors are along two adjacent sides of a parallelogram, then their resultant is equal to the corresponding diagonal in magnitude and direction

To find the magnitude of resultant vector R (+)

R = √ A² + B² + 2AB cosθ

To find the direction of resultant vector α (+)

α = tan⁻¹ { B sinθ / A+B cosθ}

Polygon law of vector addition

If a number of vectors can be represented by the sides of a polygon taken in the same order, then their resultant is represented by the closing side of the polygon taken in the opposite order.

Vector subtraction

It is the vector addition of one vector and negative vector of the other

To find the magnitude of resultant R (-)

R = √ A² + B² - 2AB cos θ

cos (180° - θ) =

- cos θ

sin (180° - θ) =

sin θ

tan (180° - θ) =

- tan θ

1 - cos θ =

2 sin² θ/2

1 + cos θ =

2 cos² θ/2

If both vectors are the same, R = (for quick answer)

2v sin θ/2

Multiplying a vector with a real number

Magnitude is changed by the factor but the direction is the same as that of the vector

Dot product

Defined as
A • B = AB cos θ

Properties of dot/scalar product (5)

• Commutative
A • B = B • A
• Dot product of perpendicular vectors is 0
• Distributive
A • (B + C) = A • B + A • C
• If A, B and C are coplanar then,
A • (B × C) = B • (C × A) = C • (A × B) = 0
• Dot product of a vector with itself gives square of its magnitude
A ⋅ A = AAcos 0°= A²

î • î =

ĵ • ĵ = k̂ • k̂ = 1

î • ĵ = ĵ • k̂ = k̂ • î =

0
î • ĵ = î ĵ cos 90° = 0

Cross product

A × B = AB sin θ n̂
where n̂ is a unit vector which points in the direction perpendicular to the plane of A & B

Properties of Cross Product (9)

• If vectors are perpendicular, then θ = 90°
A × B = AB sin 90° n̂ = AB
• Does not obey commutative rule but,
A × B = -(B × A)
• Cross product of a vector with itself is 0
A × A = 0
• Cross product of 2 antiparallel vectors is 0
A × B = AB sin 180° n̂ = 0
• Cross product of 2 parallel vectors is 0
A × B = AB sin 0° n̂ = 0
• Obey distributive law
• If 2 vectors are represented as the adjacent sides of a parallelogram then the magnitude of their cross product is the area of parallelogram
• If 2 vectors are represented as the sides of a triangle then the magnitude of their cross product is the area of triangle
• A × B = î (AᵧB𝓏 - A𝓏Bᵧ) - ĵ (AₓB𝓏 - A𝓏Bₓ) + k̂ (AₓBᵧ - AᵧBₓ)

If A = Aₓî + Aᵧĵ + A𝓏k̂ then

|A| = √ Aₓ² + Aᵧ² + A𝓏²

Vector resolution

the process of splitting a vector into its components

Rectangular components of a vector

If the components of a given vector are perpendicular to each other, then they are called rectangular components

To find components of a vector

|A| = √ Aₓ² + Aᵧ²

To find Aₓ

Aₓ = Acosθ

To find Aᵧ

Aᵧ = Asinθ

Relative velocity

It is the velocity of an object with respect to another object which is stationary or moving.

When both objects are moving in the same direction

R.v. of B w.r.t. A
Vab = Va - Vb

When B moves in opposite direction of A

R.v. of A w.r.t. B
Vba = Vb - Va

When Va and Vb are incident to each other at angle θ

Vab = √Va² + Vb² - 2VaVb cos θ

When Vb = Va

Vba = Vab = 0
Two objects seem to be stationary for one another

When Va > Vb

Vba = -Vab
Magnitude of Vba and Vab will be lower than Va and Vb. Object A appears to be faster to B and B appears slower to A

Va and Vb of opposite sign

Vba = -Vab
Magnitude of Vba and Vab will be higher than magnitude of Va and Vb. Both objects will appear faster to one another

Projectile motion

2-D motion of an object that is given an initial velocity and projected into the air at an angle. The only force acting upon the object is gravity. It follows an parabolic path determined by the effect of the initial velocity and gravitational acceleration.

Maximum height H

H = u² sin² θ / 2g
(use v² - u² = 2as)

Time of flight T

T or 2t = 2u sinθ / g
(use v = u + at)

Horizontal range R

R = u² sin2θ / g
(use s = ut + ½ at²)

For what angle θ is range maximum

45° (sin2θ = sin 90°)

Maximum range formula

Rmax = u²/g

Equation of trajectory

𝑦 = 𝑥 tanθ - ½ g𝑥²/u²cos²θ

Equation of trajectory (w.r.t. range)

𝑦 = 𝑥 tanθ (1-𝑥/R)

Angular displacement

the angle in radians through which a point or line has been rotated in a specified sense about a specified axis
θ= arc length/radius

SI unit of angular displacement

radian (rad)

SI unit of angular velocity

radian per second (rad/s)

Angular velocity

rate of change of angular displacement
ω = dθ/dt

Angular acceleration

The rate of change of angular velocity
α = dω/dt

SI unit for angular acceleration

rad/s²

Uniform circular motion

the movement of an object at a constant speed around the circumference circle

Relation between linear acceleration and angular acceleration

a = rα

Relation between linear velocity and angular velocity

v = rω

centripetal acceleration

Acc. of an object moving along the circumference of a circle and directed towards the centre.

centripetal acceleration formula

ac = v²/r

Time period

time taken for one revolution T

Frequency

/ Number of revolutions in 1 sec
v = 1/T

angular frequency

ω = 2π/T = 2πv

ac = v²/r versions

ac = Rω²
ac = R 4π² v²

centripetal

toward the center

centrifugal

away from the center