Kinematics — Motion in One and Two Dimensions
~11 min read
- Quantities: Distance (scalar), Displacement (vector). Speed (scalar), Velocity (vector). Acceleration (vector).
- Equations of motion: v = u + at, s = ut + ½at², v² = u² + 2as. For uniformly accelerated motion only.
- Projectile: Two-D motion under gravity. Range R = u²sin2θ/g. Max height H = u²sin²θ/(2g). Time of flight T = 2usinθ/g. Max range at θ=45°.
Kinematics is the description of motion without considering forces. NDA tests scalar vs vector quantities, equations of motion, free fall, and projectile motion.
Basic Quantities
| Quantity | Type | SI Unit |
|---|---|---|
| Distance | Scalar | m |
| Displacement | Vector | m |
| Speed | Scalar | m/s |
| Velocity | Vector | m/s |
| Acceleration | Vector | m/s² |
- Distance vs displacement: Distance is total path length; displacement is straight-line vector from start to end. For a closed loop, displacement = 0 but distance ≠ 0.
- Speed vs velocity: Speed has only magnitude; velocity has direction too.
Equations of Motion
For an object with uniform acceleration a:
- v = u + at
- s = ut + ½at²
- v² = u² + 2as
where u = initial velocity, v = final velocity, t = time, s = displacement.
Free fall: a = g ≈ 9.8 m/s² (downward, near Earth's surface).
Motion Graphs
| Graph | Slope means | Area under curve |
|---|---|---|
| Position-Time | Velocity | — |
| Velocity-Time | Acceleration | Displacement |
| Acceleration-Time | Jerk | Change in velocity |
Projectile Motion
Object launched at angle θ with initial speed u, under gravity g.
- Horizontal motion: constant velocity ucosθ.
- Vertical motion: usinθ initially, decelerated by g.
- Time of flight T = 2usinθ/g.
- Maximum height H = u²sin²θ/(2g).
- Range R = u²sin2θ/g.
- Maximum range when θ = 45°. R_max = u²/g.
- Trajectory: parabola.
- At highest point: vertical velocity = 0; horizontal velocity = ucosθ.
Circular Motion (Brief)
- Object moving in circle has centripetal acceleration a_c = v²/r = ω²r (directed toward centre).
- Centripetal force F_c = mv²/r.
- Tangential velocity v = ωr, where ω is angular velocity.
- For uniform circular motion: |v| constant but direction changes — hence acceleration.
NDA PYQ Examples
Q: A ball is thrown vertically up. At highest point, its acceleration is:
(a) Zero (b) g downward (c) g upward (d) Increasing
Answer: (b) g downward — gravity always acts downward.
Q: Maximum horizontal range in projectile motion is achieved at:
(a) 30° (b) 45° (c) 60° (d) 90°
Answer: (b) 45°.
Q: Distance covered by a freely falling body in 3 seconds is (g = 10):
(a) 30 m (b) 45 m (c) 60 m (d) 90 m
Answer: (b) 45 m. s = ½gt² = ½ × 10 × 9 = 45 m.
Drill Kinematics — Motion in One and Two Dimensions for NDA
NDA-pattern items on Kinematics — Motion in One and Two Dimensions with answer keys and explanations.
Start Free Mock TestFrequently Asked Questions
What is the difference between speed and velocity?
Speed is scalar (magnitude only). Velocity is vector (magnitude + direction). Average speed = total distance/time. Average velocity = displacement/time. The two can differ greatly if motion changes direction.
What is uniform motion?
Motion with constant velocity — i.e., constant speed in a straight line. Acceleration is zero. v-t graph is a horizontal line; s-t graph is a straight inclined line.
Why does a projectile follow a parabola?
Horizontal motion has constant velocity (no horizontal force). Vertical motion is uniformly accelerated (gravity). The combined trajectory — parabola — is the geometry of horizontal + uniformly accelerated motion.
What is the angle of maximum range?
45° from horizontal in vacuum (no air resistance). With air resistance, the optimal angle decreases slightly (~30-40°), depending on initial speed and drag.
Can an object have zero velocity but non-zero acceleration?
Yes. Example: a ball thrown straight up at its highest point — velocity is zero (momentarily), but gravity is still pulling it down (acceleration = g, downward).