Motion in a Plane

# Motion in a Plane

## Scalars Vs. Vectors

 Criteria Scalar Vector Definition A scalar is a quantity with magnitude only. A vector is a quantity with magnitude and direction. Direction No Yes Specified by A number (magnitude) and a unit A number (magnitude), direction and a unit Represented by quantity's symbol quantity's symbol in bold or an arrow sign above Example mass, temperature velocity, acceleration

## Position and Displacement Vectors

Position Vector: Position vector of an object at time t is the position of the object relative to the origin. It is represented by a straight line between the origin and the position at time t.

Displacement Vector: Displacement vector of an object between two points is the straight line between the two points irrespective of the path followed. The path length is always equal or greater than the displacement.

## Free and Localized Vectors

A free vector(or non-localized vector) is a vector of which only the magnitude and direction are specified, not the position or line of action. Displacing it parallel to itself leaves it unchanged.

A localized vector is a vector where line of action and position are as important as magnitude and direction. These vectors change with change in position and direction.

## Equality of Vectors

Two vectors are said to be equal only when they have same direction and magnitude.For example, two cars travelling with same speed in same direction. If they are travelling in opposite directions with same speed, then the vectors are unequal.

## Multiplication of Vectors with real numbers

 Multiplication Factor Original vector Magnitude of vector after multiplication Direction of vector after multiplication λ (>0) A λA Same as that of A -λ (<0) A λA Opposite to that of A λ (=0) A 0 (null vector) None. The initial and final positions coincide.

## Addition and Subtraction of Vectors – Triangle Method

The method of adding vectors graphically is by arranging them so that head of first is touching the tail of second vector and making a triangle by joining the open sides. This method is called head-to-tail method or triangle method of vector addition

• Commutative: A + B = B + A
• Associative: (A + B) + C = A + (B+ C)
• Adding two vectors with equal magnitudes and opposite directions results in null vector.
• Null Vector: A + (-A) = 0
• Subtraction is adding a negative vector(opposite direction) to a positive vector.
• AB = A + (-B)

## Addition of Vectors – Parallelogram Method

The method of adding vectors by parallelogram method is by:

• Touching the tail of the two vectors
• Complete a parallelogram by drawing lines from the heads of the two vectors.
• Vector resulting from the origin to the point of intersection of above lines gives the addition.

## Resolution of Vectors

A vector can be expressed in terms of other vectors in the same plane. If there are 3 vectors A, a andb, then A can be expressed as sum of a and b after multiplying them with some real numbers.

A can be resolved into two component vectors λa and μb. Hence, A = λa + μb. Here λ and μ are real numbers.

## Unit Vectors

A unit vector is a vector of unit magnitude and a particular direction.

• They specify only direction. They do not have any dimension and unit.
• In a rectangular coordinate system, the x, y and z axes are represented by unit vectors, î,ĵ andk̂
• These unit vectors are perpendicular to each other.
• |î| = |ĵ | = || = 1

In a 2-dimensional plane, a vector thus can be expressed as:

1. A = Ax î +Ay ĵ where, Ax = A cosθ and Ay = A sinθ
2. A

## Analytical Method of Vector Addition

 Vectors Sum of the vectors Subtraction of the vectors A = Ax î +Ay ĵandB = Bx î +By ĵ R = A + BR = Rx î +Ry ĵwhereRx = Ax + BxandRy = Ay + By R = A - BR = Rx î +Ry ĵwhereRx = Ax - BxandRy = Ay - By A = Ax î +Ay ĵ+Azk̂B = Bx î +By ĵ+Bzk̂ R = A + BR = Rx î +Ry ĵ+Rzk̂whereRx = Ax + BxandRy = Ay + By andRz = Az + Bz R = A - BR = Rx î +Ry ĵ+Rzk̂whereRx = Ax - BxandRy = Ay - By andRz = Az - Bz

## Quantities related to motion of an object in a plane

 Quantity Value Value in component form DisplacementΔr(Change in position) r' - r îΔx + ĵΔy Average Velocityv̅(ratio of displacement and corresponding time interval) Δr/Δt vxî + vy ĵvx= Δx/Δt, vy= Δy/Δt Instantaneous velocityv(limiting value of average velocity as the time interval approached zero) dr/dt vxî + vy ĵvx= dx/dt, vy= dy/dt Magnitude of v Direction of v, θ(direction of velocity at any point on the path is tangential to the path at that point and is in the direction of motion) tan-1(vy/vx) Average Accelerationa̅(change in velocity divided by the time interval) Δv/Δt ax î + ay ĵax= Δvx/Δt, ay= Δvy/Δt Instantaneous accelerationa(limiting value of the average acceleration as the time interval approaches zero) dv/dt ax î + ay ĵax= dvx/dt, ay= dvy/dtax= d2x/dt2, ay= d2y/dt2

## Motion in a plane with constant acceleration

Motion in a plane (two dimensions) can be treated as two separate simultaneous one-dimensional motions with constant acceleration along two perpendicular directions. X and Y directions are hence independent of each other.

If v0 being the velocity at time 0, the displacement can be written as:

x = x0 + v0xt+ ½ axt2 and y = y0 + v0yt+ ½ ayt2

 Motion of an object in a plane with constant acceleration Velocity Velocity in terms of components Displacement v = v0+ at vx = v0x + axtvy = v0y + ayt r = r0+ v0t+ ½ at2

## Relative velocity in two dimensions

The concept of relative velocity in a plane is similar to the concept of relative velocity in a straight line.

## Projectile Motion

An object that becomes airborne after it is thrown or projected is called projectile. Example, football, javelin throw, etc.

• Projectile motion comprises of two parts – horizontal motion of no acceleration and vertical motion of constant acceleration due to gravity.
• Projectile motion is in the form of a parabola, y = ax + bx2.
• Projectile motion is usually calculated by neglecting air resistance to simplify calculations.

 Quantity Value Components of velocity at time t vx = v0 cosθ0vy = v0 sinθ0–gt Position at time t x = (v0 cosθ0)ty = (v0 sinθ0)t – ½ gt2 Equation of path of projectile motion y = (tan θ0)x – gx2/2(v0 cosθ0)2 Time of maximum height tm = v0 sinθ0 /g Time of flight 2 tm = 2 (v0 sinθ0 /g) Maximum height of projectile hm = (v0 sinθ0)2/2g Horizontal range of projectile R = v02 sin 2θ0/g Maximum horizontal range (θ0=45°) Rm = v02/g

## Uniform circular motion

When an object follows a circular path at a constant speed, the motion is called uniform circular motion.

• Velocity at any point is along the tangent at that point in the direction of motion.
• Average velocity between two points is always perpendicular to Average displacement. Also, average acceleration is perpendicular to average displacement.
• For an infinitely small time interval, Δtà 0, the average acceleration becomes instantaneous acceleration which means that in uniform circular motion the acceleration of an object is always directed towards the center. This is called centripetal acceleration.

 Quantity Values Centripetal Acceleration ac = v2/R, R – radius of the circleac = ω2R, ω – angular speedac = 4π2ν2R, ν – frequency Angular Distance Δθ = ω Δt Speed v = Rω