Class 11 Physics · Chapter 3 NotesMotion in a Plane

A scalar quantity has magnitude only and is completely specified by a number with its proper unit, such as distance, mass, temperature and time. Scalars combine by ordinary algebra. A vector quantity has both magnitude and direction and obeys the triangle law or parallelogram law of addition; displacement, velocity, acceleration and force are vectors. A vector is written in bold face, such as A, and its magnitude, called its absolute value, is written as |A| or simply A. Two vectors are equal only if they have the same magnitude and the same direction; shifting a vector parallel to itself does not change it.

• Scalarmagnitude only, e.g. distance, mass, temperature, time
• Vectormagnitude and direction, e.g. displacement, velocity, acceleration, force
• A = B only if |A| = |B| and their directions are the same

To locate an object moving in a plane, we choose an origin O. The position vector r of a point P is the vector OP, and in component form r = x i + y j, where x and y are the coordinates of the object. If the object moves from P at time t to P′ at time t′, the displacement vector is Δr = r′ − r, directed from P to P′. In components, Δr = Δx i + Δy j, where Δx = x′ − x and Δy = y′ − y. The displacement depends only on the initial and final positions, not on the actual path followed, so its magnitude is either less than or equal to the path length.

• Position vectorr = x i + y j
• DisplacementΔr = r′ − r = Δx i + Δy j
• Magnitude of displacement ≤ path length

Multiplying a vector A by a positive number λ gives a vector λA whose magnitude is λ times |A| and whose direction is the same as that of A. If λ is negative, the direction of λA is opposite to that of A. If the scalar has its own physical dimension, the dimension of λA is the product of the dimensions of λ and A; for example, multiplying a velocity vector by a time interval gives a displacement vector. Adding a vector to its negative gives the null vector 0, whose magnitude is zero and whose direction cannot be specified. Also, A + 0 = A, λ0 = 0 and 0A = 0.

• |λA| = λ|A| for λ > 0; direction reverses for λ < 0
• A + (−A) = 0, the null vector with |0| = 0
• A + 0 = A, λ0 = 0, 0A = 0

In the head-to-tail (triangle) method, vector B is placed with its tail at the head of A, and the vector drawn from the tail of A to the head of B gives the resultant R = A + B. In the parallelogram method, A and B are drawn from a common origin and the diagonal of the completed parallelogram gives the resultant; the two methods are equivalent. Vector addition is commutative, A + B = B + A, and associative, (A + B) + C = A + (B + C). Subtraction is defined as A − B = A + (−B), where −B has the same magnitude as B but the opposite direction.

• Triangle lawplace vectors head to tail; resultant joins first tail to last head
• Parallelogram lawresultant is the diagonal from the common origin
• A + B = B + A; (A + B) + C = A + (B + C)
• A − B = A + (−B)

Any vector A in a plane can be expressed as the sum of two component vectors along two non-collinear directions a and b: A = λa + μb, where λ and μ are real numbers. It is most convenient to resolve a vector along the axes of a rectangular coordinate system using unit vectors. A unit vector has magnitude 1 and no dimension or unit; i, j and k point along the x-, y- and z-axes. A vector A in the x-y plane is written as A = Ax i + Ay j, where Ax = A cos θ and Ay = A sin θ if A makes an angle θ with the x-axis. Conversely, A = √(Ax² + Ay²) and tan θ = Ay/Ax. A component can be positive, negative or zero depending on θ.

• A = Ax i + Ay j, with Ax = A cos θ and Ay = A sin θ
• A = √(Ax² + Ay²), θ = tan⁻¹(Ay/Ax)
• Unit vectors i, j, k have magnitude 1 and give direction only

Graphical addition is useful for visualising vectors but can be tedious and limited in accuracy. In the analytical method, vectors are added by combining their components. If A = Ax i + Ay j and B = Bx i + By j, then R = A + B = Rx i + Ry j, where Rx = Ax + Bx and Ry = Ay + By. Each component of the resultant is the sum of the corresponding components of the vectors being added. The same idea extends to three dimensions and to the addition or subtraction of any number of vectors; for example, if T = a + b − c, then Tx = ax + bx − cx, and similarly for the y and z components.

• R = A + B gives Rx = Ax + Bx and Ry = Ay + By
• Extends to three dimensionsRz = Az + Bz
• For T = a + b − cTx = ax + bx − cx, and similarly for y and z

For an object moving in a plane with constant acceleration a, the velocity and position at time t are given by v = v₀ + at and r = r₀ + v₀t + ½at², where v₀ and r₀ are the velocity and position at t = 0. In component form, vx = vox + ax t, vy = voy + ay t, x = x₀ + vox t + ½ax t² and y = y₀ + voy t + ½ay t². This shows that motion in a plane can be treated as two separate simultaneous one-dimensional motions with constant acceleration along two perpendicular directions. The choice of perpendicular directions is convenient in many physical situations, such as projectile motion.

• v = v₀ + at and r = r₀ + v₀t + ½at²
• x = x₀ + vox t + ½ax t²; y = y₀ + voy t + ½ay t²
• Motion in a plane = two independent one-dimensional motions

An object in flight after being thrown or projected is called a projectile. Neglecting air resistance, its acceleration is that due to gravity, directed vertically downward, so ax = 0 and ay = −g. If it is projected with speed v₀ at angle θ₀ to the x-axis from the origin, then x = (v₀ cos θ₀)t and y = (v₀ sin θ₀)t − ½gt², while vx = v₀ cos θ₀ and vy = v₀ sin θ₀ − gt. The horizontal component of velocity stays constant and the vertical component changes like free fall. Eliminating t gives the path y = (tan θ₀)x − gx²/(2(v₀ cos θ₀)²), which is a parabola. The time to reach maximum height is tm = v₀ sin θ₀/g, the maximum height is hm = (v₀ sin θ₀)²/(2g), the time of flight is Tf = 2v₀ sin θ₀/g, and the horizontal range is R = v₀² sin 2θ₀/g, which is maximum when θ₀ = 45°.

• x = (v₀ cos θ₀)t; y = (v₀ sin θ₀)t − ½gt²
• Path is a parabolay = (tan θ₀)x − gx²/(2(v₀ cos θ₀)²)
• Maximum height hm = (v₀ sin θ₀)²/(2g)
• Time of flight Tf = 2v₀ sin θ₀/g
• Range R = v₀² sin 2θ₀/g, maximum at θ₀ = 45°

When an object moves along a circular path at constant speed, its motion is called uniform circular motion. Although the speed is constant, the direction of velocity changes continuously, so the object accelerates. The velocity at any point is along the tangent to the circle, and the acceleration is directed towards the centre, which is why it is called centripetal acceleration. Its magnitude is ac = v²/R, where v is the speed and R is the radius. Angular speed ω is the time rate of change of angular displacement, and v = ωR, so ac = ω²R. If T is the time period and ν the frequency, then ω = 2πν, v = 2πνR and ac = 4π²ν²R. The magnitude of the centripetal acceleration is constant, but its direction changes, so it is not a constant vector.

• Velocity is tangential; acceleration is directed towards the centre
• ac = v²/R = ω²R
• v = ωR; ω = 2πν; v = 2πνR; ac = 4π²ν²R
• Centripetal acceleration is not a constant vector