Kinematics Motion 1D 2D Questions for AGNIVEER VAYU

Equations of Motion (1D)

These three equations describe how objects move in a straight line under constant acceleration. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Think of them as the 'holy trinity' of kinematics—memorize all three forms because different problems give you different known quantities. For example, if you don't know time, use the equation without t. If you don't know final velocity, use the one without v.

• v = u + at (velocity changes linearly with time)
• s = ut + (1/2)at^2 (displacement includes both initial motion and acceleration effect)
• v^2 = u^2 + 2as (relates velocity and displacement without time)
• Average velocity = (u + v)/2 when acceleration is constant
• Displacement is area under velocity-time graph
• Use sign convention: take one direction as positive, opposite as negative

Projectile Motion (2D)

Projectile motion is 1D kinematics applied independently to two perpendicular directions: horizontal (x) and vertical (y). The key insight is that horizontal and vertical motions are completely independent—there is no horizontal acceleration (ignoring air resistance), so horizontal velocity stays constant. Vertical motion is just free fall under gravity. Decompose the initial velocity into components using trigonometry, then treat each direction separately.

• Horizontal: x = u_x * t (constant velocity, no acceleration)
• Vertical: y = u_y * t - (1/2)gt^2 (free fall under gravity g = 10 m/s^2)
• Initial velocity components: u_x = u*cos(theta), u_y = u*sin(theta)
• At maximum height: v_y = 0 (vertical velocity becomes zero momentarily)
• Time of flight: total time in air = 2*u_y/g = 2*u*sin(theta)/g
• Range (horizontal distance): R = u^2*sin(2*theta)/g (maximum at 45 degrees)

Relative Motion

Relative motion asks: how does object A move as seen from object B's perspective? The trick is vector subtraction. Velocity of A relative to B is v_AB = v_A - v_B (as vectors). This is crucial for river-crossing problems, rain problems, and aircraft problems in Agniveer exams. Always draw a vector diagram—it clarifies the geometry instantly. Remember: relative velocity is the difference of velocities, not the difference of speeds.

• Relative velocity: v_AB = v_A - v_B (vector subtraction)
• In river problems: boat velocity relative to ground = boat velocity relative to water + water velocity relative to ground
• For perpendicular motion: use Pythagoras to find resultant
• Relative displacement: s_AB = s_A - s_B
• Time to collision/meeting: use relative velocity and relative displacement
• Rain appears to come at an angle due to observer's motion—use velocity triangle

Graphs in Kinematics

Velocity-time (v-t) and displacement-time (s-t) graphs are visual shortcuts in Agniveer exams. A v-t graph's slope is acceleration. The area under a v-t graph is displacement. An s-t graph's slope is velocity. A curved s-t graph means acceleration is present. These graphs often appear as 'read the graph and answer' questions—practice interpreting them quickly.

• Slope of s-t graph = velocity
• Slope of v-t graph = acceleration
• Area under v-t graph = displacement
• Straight line in s-t graph means constant velocity
• Curved s-t graph (parabola) means constant acceleration
• Horizontal line in v-t graph means zero acceleration (uniform motion)

• Forgetting to resolve velocity into components in 2D problems—always split u into u*cos(theta) and u*sin(theta) at the start
• Using g = 9.8 m/s^2 when exam expects g = 10 m/s^2—check the question; most Agniveer papers use 10 for simplicity
• Mixing up 'range' with 'maximum height'—range is horizontal distance, height is vertical; they use different formulas
• In relative motion, adding velocities instead of subtracting—v_AB = v_A - v_B, not v_A + v_B
• Ignoring sign convention—treating upward and downward as same sign leads to wrong answers in vertical motion

• SUVAT: S=displacement, U=initial velocity, V=final velocity, A=acceleration, T=time. Pick the three you know, find the two you need.
• For projectile: 'Horizontal is lazy, vertical is crazy'—horizontal velocity never changes, vertical velocity changes every instant due to gravity.
• Range formula sin(2*theta): Remember 2*theta because sin(2*theta) = 2*sin(theta)*cos(theta), which is why 45 degrees gives maximum range (sin(90) = 1).
• Relative velocity: 'Subtract to see'—v_AB = v_A minus v_B. Think of it as 'what A looks like from B's frame'.

• Agniveer Vayu typically gives 1-2 projectile motion questions per paper—focus on range, time of flight, and maximum height calculations. Recent papers favor 30-45-60 degree angles.
• Relative motion appears in 'boat crossing river' or 'rain angle' format—always draw a velocity triangle; it's faster than algebra.
• Graph-based questions are common—practice reading v-t and s-t graphs in 30 seconds. Examiners test if you can extract slope and area quickly.
• Numerical answers are usually 'nice' numbers (10, 20, 25, 50 m, etc.)—if your answer is messy, recheck your calculation or g value.
• Time management: kinematics questions are solvable in 2-3 minutes if you know which equation to use. Spend 10 seconds choosing the right formula, then calculate.