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| Step-by-step solution of particle motion in x-y plane using velocity and acceleration vectors. |
Example 3.5 — Easy Beginner Notes
A particle starts from the origin at t = 0 with velocity:
v = 5i m/s
Constant acceleration is:
a = (3i + 2j) m/s²
- Acceleration in x-direction = 3 m/s²
- Acceleration in y-direction = 2 m/s²
Step 1: Position Formula
r = r₀ + v₀t + ½at²
Since particle starts from origin:
r₀ = 0
So,
r = v₀t + ½at²
Step 2: Put Given Values
v₀ = 5i
a = 3i + 2j
Substitute in position formula:
r = 5it + ½(3i + 2j)t²
Step 3: Simplify
r = 5ti + 1.5t²i + 1.0t²j
r = (5t + 1.5t²)i + (1.0t²)j
Step 4: Write x and y Coordinates
x = 5t + 1.5t²
y = 1.0t²
Part (a): Find y-coordinate when x = 84 m
Given:
x = 84
Put in x-equation:
5t + 1.5t² = 84
Solve the equation:
1.5t² + 5t - 84 = 0
t = 6 s
1.5t² + 5t - 84 = 0
t = 6 s
Step 5: Find y-coordinate
y = 1.0t²
Put t = 6:
y = 1.0 × (6)²
y = 36 m
Answer (a): y = 36 m
Part (b): Find Speed at t = 6 s
Step 6: Velocity Formula
v = v₀ + at
Substitute values:
v = 5i + (3i + 2j)t
Put t = 6 s:
v = 5i + (3×6)i + (2×6)j
v = 23i + 12j
- x-component of velocity = 23 m/s
- y-component of velocity = 12 m/s
Step 7: Find Speed
Speed = √(vx² + vy²)
Speed = √(23² + 12²)
Speed = √(529 + 144)
Speed = √673
Speed ≈ 26 m/s
Answer (b): Speed = 26 m/s
Quick Beginner Understanding
- i represents x-direction
- j represents y-direction
- Position formula gives location of particle
- Velocity formula gives motion speed and direction
- Speed is magnitude of velocity vector
- x and y motions are solved separately
Internal Links
Introduction to Motion in a Plane
Vector Basics for Beginners
Velocity and Acceleration Explained
Kinematics Formulas with Examples
Solved Numerical Problems for Class 11 Physics
Projectile Motion Notes
NCERT Physics Chapter 3 Solutions
Scalar and Vector Quantities
Motion Along Straight Line Notes
Physics Formula Sheet for Students
