Motion in a Plane
Vector addition, resolution, and independence of perpendicular motions. (Physics › Kinematics, NEET UG syllabus.)
What is Motion in a Plane?
A physical quantity having both magnitude and direction (e.g., displacement, velocity, acceleration, force).
Key formula / rule: Vector Resolution (x-component)
Key points
- Define and differentiate between scalar and vector quantities.
- Perform vector addition and subtraction using analytical and graphical methods.
- Resolve a vector into its rectangular components.
- Apply the principle of independence of perpendicular motions to solve 2D problems.
Common exam trap
Confusing scalar and vector quantities.
Definitions
- Term
Vector
- Meaning
A physical quantity having both magnitude and direction (e.g., displacement, velocity, acceleration, force).
- Term
Scalar
- Meaning
A physical quantity having only magnitude (e.g., distance, speed, mass, time, temperature).
- Term
Projectile
- Meaning
An object given an initial velocity and then allowed to move freely under the influence of gravity alone.
- Term
Trajectory
- Meaning
The path followed by a projectile or any moving object.
- Term
Relative Velocity
- Meaning
The velocity of an object as observed from another moving object or frame of reference.
- Term
Resolution of Vector
- Meaning
The process of splitting a vector into two or more components along specified directions, typically perpendicular axes.
Learning objectives
Define and differentiate between scalar and vector quantities.
Perform vector addition and subtraction using analytical and graphical methods.
Resolve a vector into its rectangular components.
Apply the principle of independence of perpendicular motions to solve 2D problems.
Analyze and solve problems related to projectile motion (time of flight, maximum height, range).
Understand and apply the concept of relative velocity in two dimensions.
Derive the equations for projectile motion.
Formulae
- Name
Vector Resolution (x-component)
- Note
θ is the angle with the positive x-axis.
- Expression
Ax = A cosθ
- Name
Vector Resolution (y-component)
- Note
θ is the angle with the positive x-axis.
- Expression
Ay = A sinθ
- Name
Magnitude of Resultant Vector
- Note
Where Rx and Ry are the components of the resultant vector.
- Expression
R = √(Rx^2 + Ry^2)
- Name
Direction of Resultant Vector
- Note
α is the angle of the resultant with the positive x-axis.
- Expression
tanα = Ry / Rx
- Name
Projectile Motion - Time of Flight
- Note
u is initial speed, θ is angle of projection with horizontal.
- Expression
T = (2u sinθ) / g
- Name
Projectile Motion - Maximum Height
- Note
u is initial speed, θ is angle of projection with horizontal.
- Expression
H = (u2 sin2θ) / (2g)
- Name
Projectile Motion - Horizontal Range
- Note
u is initial speed, θ is angle of projection with horizontal.
- Expression
R = (u2 sin(2θ)) / g
- Name
Equation of Trajectory
- Note
Relates vertical displacement (y) to horizontal displacement (x).
- Expression
y = x tanθ - (g x2) / (2 u2 cos2θ)
- Name
Relative Velocity (Vector Form)
- Note
Velocity of A relative to B. All are vectors.
- Expression
VAB = VA - VB
Prerequisites
Basic understanding of scalars and vectors.
Vector addition and subtraction (graphical and analytical methods).
Resolution of vectors into components.
Basic trigonometry (sine, cosine, tangent).
One-dimensional kinematics (equations of motion under constant acceleration).
Concept of displacement, velocity, and acceleration.
Common mistakes
Confusing scalar and vector quantities.
Incorrectly resolving vectors into components (e.g., using sine instead of cosine).
Applying 1D kinematic equations without resolving components first.
Assuming horizontal acceleration in projectile motion (it's zero, neglecting air resistance).
Incorrectly applying sign conventions for displacement, velocity, and acceleration in vertical motion.
Forgetting that 'g' is always downwards, regardless of the direction of vertical motion.
Not understanding the difference between speed and velocity.
Incorrectly performing vector addition/subtraction, especially for relative velocity problems.
Keywords
Vector
Scalar
Projectile Motion
Trajectory
Range
Height
Time of Flight
Relative Velocity
Resolution of Vectors
Independence of Motion
Parabolic Path
Kinematics
Practice preview
A particle has a velocity of 3 m/s towards east and 4 m/s towards north. What is the magnitude of its resultant velocity?…
easy
Which of the following statements is true regarding the horizontal motion of a projectile launched in the absence of air resistance?…
easy
A boatman wants to cross a river of width 500 m flowing at 5 m/s. He can row his boat at 10 m/s in still water. If he rows perpendicular to the river flow, what is the time taken to cross the river?…
medium