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Speed, Velocity & Acceleration in Physics: Kinematics & Motion Guide
Kinematics is the branch of classical mechanics that describes the motion of points, bodies, and systems of objects without considering the forces that cause the motion. Master the fundamental distinction between scalar distance and vector displacement, speed and velocity, uniform acceleration, and projectile flight trajectories.
This 4,000+ word comprehensive exam guide covers **Distance vs Displacement**, **Speed vs Velocity**, **Average Speed Formulas ($v_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2}$)**, **Acceleration ($a = \frac{v-u}{t}$)**, **The Three Fundamental Equations of Motion ($v = u + at$, $s = ut + \frac{1}{2}at^2$, $v^2 = u^2 + 2as$)**, **Motion Under Gravity ($g = 9.8\text{ m/s}^2$)**, **Projectile Motion Parameters ($T, H, R$)**, **Uniform Circular Motion**, and solved numerical problems for SSC CGL, RRB NTPC, and UPSC Prelims.
Table of Contents
- 1. Distance vs Displacement: Definitions & Key Differences
- 2. Speed vs Velocity & Average Speed Formulas
- 3. Acceleration: Uniform, Non-Uniform & Retardation
- 4. The Three Fundamental Equations of Motion (Derivation & Use)
- 5. Motion Under Gravity & Free Fall Physics ($g = 9.81\text{ m/s}^2$)
- 6. Projectile Motion: Time of Flight ($T$), Max Height ($H$) & Range ($R$)
- 7. Uniform Circular Motion & Centripetal Acceleration ($a_c = v^2/r$)
- 8. Kinematic Motion Graphs ($s-t$, $v-t$, $a-t$)
- 9. Solved Numerical Examples for Competitive Exams
- 10. Must Remember Points for Quick Revision
- 11. Frequently Asked Questions (FAQ)
Key Takeaways & Core Highlights
- Distance vs Displacement: Distance is total path length (scalar, always $\ge 0$). Displacement is shortest straight line from initial to final position (vector, can be positive, negative, or zero!). $\text{Displacement} \le \text{Distance}$.
- Average Speed (Equal Distances): $v_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2}$ (Harmonic Mean).
- Average Speed (Equal Time Intervals): $v_{\text{avg}} = \frac{v_1 + v_2}{2}$ (Arithmetic Mean).
- The 3 Equations of Motion (Uniform Acceleration $a$):
- $v = u + a t$
- $s = u t + \frac{1}{2} a t^2$
- $v^2 = u^2 + 2 a s$
- Free Fall under Gravity ($g = 9.8\text{ m/s}^2$): Drop from rest ($u=0$): $v = g t$, $h = \frac{1}{2} g t^2$, $v^2 = 2 g h$. Time to reach max height $t = u/g$.
- Projectile Motion ($45^\circ$ Maximum Range): Maximum range occurs at projection angle **$\theta = 45^\circ$** ($R_{\text{max}} = u^2/g$). Range is identical for complementary angles $\theta$ and $(90^\circ - \theta)$!
1. Distance vs Displacement: Definitions & Key Differences
| Property | Distance ($s$) | Displacement ($\vec{s}$) |
|---|---|---|
| Definition | Total length of actual path traversed by moving body | Shortest straight-line distance from initial to final position |
| Quantity Type | Scalar (Magnitude only) | Vector (Magnitude and specific direction) |
| Value Constraints | Always positive ($>0$) or zero (at rest) | Can be positive, negative, or zero |
| Path Dependence | Depends on exact path taken | Independent of path taken (depends only on end points) |
2. Speed vs Velocity & Average Speed Formulas
1. Average Speed (Equal Distances):
If a body travels distance $d$ at speed $v_1$ and another equal distance $d$ at speed $v_2$:
$$v_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2}$$2. Average Speed (Equal Times):
If a body travels for time $t$ at speed $v_1$ and another equal time $t$ at speed $v_2$:
$$v_{\text{avg}} = \frac{v_1 + v_2}{2}$$4. The Three Fundamental Equations of Motion
$$\text{1st Equation: } v = u + a \cdot t$$ $$\text{2nd Equation: } s = u \cdot t + \frac{1}{2} a \cdot t^2$$ $$\text{3rd Equation: } v^2 = u^2 + 2 a \cdot s$$ $$\text{Distance in } n^{\text{th}} \text{ second: } s_n = u + \frac{a}{2}(2n - 1)$$6. Projectile Motion: Time of Flight ($T$), Max Height ($H$) & Range ($R$)
When an object is thrown into the air at an angle $\theta$ to the horizontal with initial velocity $u$:
$$\text{Time of Flight: } T = \frac{2 u \sin\theta}{g}$$ $$\text{Maximum Height: } H = \frac{u^2 \sin^2\theta}{2 g}$$ $$\text{Horizontal Range: } R = \frac{u^2 \sin(2\theta)}{g}$$Important Projectile Properties:
- Maximum horizontal range $R_{\text{max}} = \frac{u^2}{g}$ is achieved at projection angle **$\theta = 45^\circ$**.
- Horizontal range is IDENTICAL for complementary angles of projection: $\theta$ and $(90^\circ - \theta)$ (e.g. $30^\circ$ and $60^\circ$ yield the exact same range $R$).
9. Solved Numerical Examples for Competitive Exams
Numerical Problem 1 (Average Speed Calculation):
Question: A car covers the first half of a journey at $40\text{ km/h}$ and the second half of the journey at $60\text{ km/h}$. Calculate its average speed.
Solution:
$$v_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 \times 40 \times 60}{40 + 60} = \frac{4800}{100} = 48 \text{ km/h}$$Answer: Average speed is $48\text{ km/h}$.
Numerical Problem 2 (Free Fall Distance):
Question: A stone dropped from the top of a tower reaches the ground in $4\text{ seconds}$. Calculate the height of the tower ($g = 10\text{ m/s}^2$).
Solution:
$$h = u t + \frac{1}{2} g t^2 = (0 \times 4) + \frac{1}{2} \times 10 \times (4)^2 = 5 \times 16 = 80 \text{ meters}$$Answer: Height of the tower is $80\text{ meters}$.
10. Must Remember Points for Quick Revision
Exam Revision Cheat Sheet:
- Distance vs Displacement: Displacement can be 0 even if distance is positive!
- Average Speed (Equal Distances): $v_{\text{avg}} = 2 v_1 v_2 / (v_1 + v_2)$.
- 3 Motion Equations: $v = u + at$, $s = ut + \frac{1}{2}at^2$, $v^2 = u^2 + 2as$.
- Free Fall ($u=0$): $v = gt$, $h = \frac{1}{2}gt^2$, $v^2 = 2gh$.
- Projectile Max Range: At $\theta = 45^\circ$. Equal range for $\theta$ and $90^\circ - \theta$.
- Centripetal Acceleration: $a_c = v^2/r = \omega^2 r$. Directed toward center.
11. Frequently Asked Questions (FAQ)
What is the fundamental difference between Distance and Displacement?
Distance is the total path length traveled by a body regardless of direction. It is a scalar quantity and is always positive or zero. Displacement is the shortest straight-line vector distance from the initial position to the final position. It is a vector quantity and can be positive, negative, or zero.
What are the three fundamental equations of motion under uniform acceleration?
1. First Equation (Velocity-Time): v = u + a · t, 2. Second Equation (Position-Time): s = u · t + ½ a · t², 3. Third Equation (Position-Velocity): v² = u² + 2 a · s.
What is the average speed formula when a body covers equal distances at speeds v₁ and v₂?
When equal distances are covered at speeds v₁ and v₂, the average speed is given by the Harmonic Mean: v_avg = (2 v₁ v₂) / (v₁ + v₂).
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