Chapter 3: Motion in a Plane | Summary

Multiplying a vector A by a positive scalar λ gives a vector in the same direction as A but with magnitude λ|A|. Multiplying by a negative scalar −λ gives a vector in the opposite direction with magnitude λ|A|. When λ has its own physical dimension, the product λA has dimensions equal to the product of the dimensions of λ and A. For example, multiplying a velocity vector by time gives a displacement vector.

The null vector (zero vector) 0 results when a vector is added to its negative, or when any vector is multiplied by zero. The null vector has zero magnitude and no defined direction. Its key properties are: A + 0 = A, λ0 = 0, and 0A = 0.

The head-to-tail method (triangle method) adds vectors by placing the tail of B at the head of A; the resultant R = A + B is the vector from the tail of A to the head of B. Vector addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)). The parallelogram method brings both vectors to a common origin; the resultant is the diagonal of the completed parallelogram.

Vector subtraction is defined as A − B = A + (−B), where −B has the same magnitude as B but opposite direction. Both graphical methods — triangle and parallelogram — yield equivalent results. The magnitude of displacement is always less than or equal to the path length between two points.

Any vector A in a plane can be resolved into two component vectors along any two non-collinear vectors a and b in that plane: A = λa + μb, where λ and μ are real numbers. The most convenient resolution uses unit vectors along coordinate axes. Unit vectors i-hat, j-hat, and k-hat have magnitude 1 and point along the x-, y-, and z-axes respectively, with |i-hat| = |j-hat| = |k-hat| = 1.

A vector A in the x-y plane is expressed as A = Ax·i-hat + Ay·j-hat, where Ax = A cos θ and Ay = A sin θ, θ being the angle with the x-axis. The magnitude is A = √(Ax² + Ay²) and the direction is given by tan θ = Ay/Ax. In three dimensions, A = Ax·i-hat + Ay·j-hat + Az·k-hat, with A = √(Ax² + Ay² + Az²).

The analytical method adds vectors component-wise. If R = A + B, then Rx = Ax + Bx and Ry = Ay + By. This method is more accurate and less tedious than the graphical approach. The law of cosines gives the magnitude: R = √(A² + B² + 2AB cos θ), and the law of sines gives the direction. These results are derived from the parallelogram geometry.

The analytical method extends to any number of vectors and to three dimensions. For vectors a, b, c with components along x, y, z, the resultant T = a + b − c has components Tx = ax + bx − cx, Ty = ay + by − cy, and Tz = az + bz − cz.

The position vector of a particle in the x-y plane is r = x·i-hat + y·j-hat. Displacement is Δr = r′ − r = Δx·i-hat + Δy·j-hat. Average velocity is v-bar = Δr/Δt, and instantaneous velocity v = dr/dt, which is always directed tangent to the path. The velocity components are vx = dx/dt and vy = dy/dt; the magnitude is v = √(vx² + vy²).

Average acceleration is a-bar = Δv/Δt, and instantaneous acceleration a = dv/dt, with components ax = dvx/dt and ay = dvy/dt. In two or three dimensions, velocity and acceleration can make any angle between 0° and 180°, unlike one-dimensional motion where they must be collinear.

For constant acceleration a in a plane, starting with velocity v0 at t = 0, the velocity at time t is v = v0 + at, and the position is r = r0 + v0t + (1/2)at². In component form: vx = vox + axt, vy = voy + ayt, x = x0 + voxt + (1/2)axt², y = y0 + voyt + (1/2)ayt².

A key result is that motions along the x- and y-directions are independent of each other. Two-dimensional motion with constant acceleration can therefore be treated as two simultaneous one-dimensional motions along perpendicular directions. This principle is fundamental to analysing projectile motion.

A projectile is an object in flight after being projected. Its motion combines uniform horizontal motion (no acceleration) and free-fall vertical motion (acceleration g downward). The horizontal component of velocity remains constant throughout: vx = v0 cos θ0. The vertical component changes as vy = v0 sin θ0 − gt. Galileo first stated this independence of horizontal and vertical components in 1632.

The trajectory of a projectile is a parabola given by y = (tan θ0)x − gx²/[2(v0 cos θ0)²]. The maximum height is hm = (v0 sin θ0)²/(2g), time to reach it is tm = v0 sin θ0/g, and total time of flight is Tf = 2v0 sin θ0/g. The horizontal range is R = v0² sin 2θ0/g, which is maximum at θ0 = 45°, giving Rm = v0²/g.

Uniform circular motion is motion at constant speed along a circular path. Although the speed is constant, the velocity direction changes continuously, so the object undergoes acceleration. This centripetal acceleration is always directed towards the centre of the circle, with magnitude ac = v²/R. It is not a constant vector because its direction continuously changes.

Angular speed ω is the rate of change of angular displacement: ω = Δθ/Δt, with v = Rω. Centripetal acceleration can be expressed as ac = ω²R. For an object with time period T and frequency ν: ω = 2πν, v = 2πRν, and ac = 4π²ν²R. Centripetal acceleration was first thoroughly analysed by Christiaan Huygens in 1673.

Chapter 3 establishes the mathematical framework — vectors and their operations — required to describe motion in two and three dimensions. The key insight is that two-dimensional motion with constant acceleration decomposes into two independent one-dimensional motions along perpendicular axes. This decomposition elegantly explains projectile motion as the superposition of uniform horizontal motion and uniformly accelerated vertical motion, yielding a parabolic trajectory. Uniform circular motion, despite constant speed, involves continuous centripetal acceleration directed towards the centre.

For board examinations, this chapter is highly important. Students must be proficient in vector addition (both graphical and analytical), resolving vectors into components, deriving and applying projectile motion equations (range, maximum height, time of flight), and understanding centripetal acceleration. Numerical problems on projectile motion and circular motion are standard in CBSE examinations and require a clear grasp of the independence of perpendicular components.