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MOTION IN TWO
DIMENSIONS
● A projectile is a body that is in flight after being projected and follows a path determined entirely by the effects of gravitational acceleration and air resistance. Assumptions of Projectile Motion ● There is no resistance due to air ● The effect due to curvature of earth is negligible ● The effect due to rotation of earth is negligible ● For all points of the trajectory, the acceleration due to gravity ‘g’ is constant in magnitude and direction Types of Projectile Motion ● Oblique projectile motion ● Horizontal projectile motion ● Projectile motion on an inclined plane ● Oblique projectile motion (y = ax – bx^2 ) Displacement of projectile :
● The angle of elevation φ of the highest point of the projectile and the angle of projection θ are related to each other as Instantaneous velocity: Change in velocity: Change in momentum: Between projection point and highest point For the complete projectile motion Time of flight: ● The total time taken by the projectile to go up and come down to the same level from which it was projected is called time of flight. Time of Flight (tf) = Time of Ascent + Time of Ascent Horizontal range: ● It is the horizontal distance travelled by a body during the time of flight. So by using second equation of motion in x-direction
; (when sin^2 θ = max = 1 i.e., θ = 90o) For complementary angles of projection θ and 90o^ – θ Ratio of maximum height Projectile passing through two different points on same height at time t 1 and t 2 : Height (y): Time (t 1 and t 2 ): Motion of a projectile as observed from another projectile: ● Suppose two balls A and B are projected simultaneously from the origin, with initial velocities u 1 and u 2 at angle θ 1 and θ 2 , respectively with the horizontal.
● The instantaneous positions of the two balls are given by Ball A: x 1 = (u 1 cosθ 1 )t, Ball B: x 2 = (u 2 cosθ 2 )t, Energy of projectile: ● When a projectile moves upward its kinetic energy decreases, potential energy increases but the total energy always remains constant. Kinetic energy ∴ Potential energy Total energy = Kinetic energy + Potential energy
Direction of instantaneous velocity: Time of flight: Horizontal range: ● If projectiles A and B are projected horizontally with different initial velocity from same height and third particle C is dropped from same point then o All three particles will take equal time to reach the ground. o Their net velocity would be different but all three particles possess same vertical component of velocity. o The trajectory of projectiles A and B will be straight line w.r.t. particle C. ● If various particles thrown with same initial velocity but in different direction then
o They strike the ground with same speed at different times irrespective of their initial direction of velocities. o Time would be least for particle E which was thrown vertically downward. o Time would be maximum for particle A which was thrown vertically upward. Projectile Motion on an Inclined Plane ● Let a particle be projected up with a speed u from an inclined plane which makes an angle with the horizontal and velocity of projection makes an angle θ with the inclined plane. Time of flight: Time of flight on an inclined plane Maximum height: Maximum height on an inclined plane Horizontal range: Horizontal range on an inclined plane
body it must be given some initial velocity and a force must then act on the body which is always directed at right angles to instantaneous velocity. Displacement: ● The change of position vector or the displacement of the particle from position A to the position B is given by referring the figure. Putting we obtain Distance: ● The distanced covered by the particle during the time t is given as d = length of the arc AB = r θ Ratio of distance and displacement:
Angular displacement (θ): ● The angle turned by a body moving in a circle from some reference line is called angular displacement. ● Dimension = [M^0 L^0 T^0 ] (as θ =. ● Units = Radian or Degree. It is some time also specified in terms of fraction or multiple of revolution. ● ● Angular displacement is a axial vector quantity. ● Its direction depends upon the sense of rotation of the object and can be given by Right Hand Rule; which states that if the curvature of the fingers of right hand represents the sense of rotation of the object, then the thumb, held perpendicular to the curvature of the fingers, represents the direction of angular displacement vector. ● Relation between linear displacement and angular displacement or Angular velocity (ω): (i) Angular velocity of an object in circular motion is defined as the time rate of change of its angular displacement. Angular velocity ω = (ii) Dimension: [M^0 L^0 T–1] (iii) Units: Radians per second (rad.s–1) or Degree per second. (iv) Relation between angular velocity and linear velocity (v) For uniform circular motion ω remains constant where as for non-uniform motion ω varies with respect to time. (vi) Nothing actually moves in the direction of the angular velocity vector. The direction of simply represents that the circular motion is taking place in a plane perpendicular to it. (Pseudo Vectors / Axial Vectors)
● The time taken by one to complete one revolution around O with respect to the other (i.e., time in which B complete one revolution around O with respect to the other (i.e., time in which B completes one more or less revolution around O than A) Special case : If and so T = ∞., particles will maintain their position relative to each other. This is what actually happens in case of geostationary satellite (ω 1 = ω 2 = constant) Angular acceleration (α): ● Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity. ● If Δω be the change in angular velocity of the object in time interval Δt, while moving on a circular path, then angular acceleration of the object will be ● Units: rad. s– ● Dimension: [M^0 L^0 T–2] ● Relation between linear acceleration and angular acceleration ● For uniform circular motion since ω is constant so ● For non-uniform circular motion Centripetal Acceleration ● Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.
● It always acts on the object along the radius towards the centre of the circular path. Magnitude of centripetal acceleration , Direction of centripetal acceleration: ● The centripetal acceleration vector acts along the radius of the circular path at that point and is directed towards the centre of the circular path. Non-Uniform Circular Motion ● If the speed of the particle in a horizontal circular motion changes with respect to time, then its motion is said to be non-uniform circular motion. (Angular acceleration) (Linear velocity) Thus the resultant acceleration of the particle at P has two component accelerations Tangential acceleration: ● It acts along the tangent to the circular path at P in the plane of circular path. According to right hand rule since and are perpendicular to each other, therefore, the magnitude of tangential acceleration is given by
● Centripetal force is that force which is required to move a body in a circular path with uniform speed. The force acts on the body along the radius and towards centre. Formulae for centripetal force: Centripetal force in different situation Situation Centripetal Force A particle tied to a string and whirled in a horizontal circle Tension in the string Vehicle taking a turn on a level road Frictional force exerted by the road on the tyres A vehicle on a speed breaker Weight of the body or a component of weight Revolution of earth around the sun Gravitational force exerted by the sun Electron revolving around the nucleus in an atom Coulomb attraction exerted by the protons in the nucleus A charged particle describing a circular path in a magnetic field Magnetic force exerted by the agent that sets up the magnetic field Work Done by Centripetal Force ● The work done by centripetal force is always zero as it is perpendicular to velocity and hence instantaneous displacement. Work done = Increment in kinetic energy of revolving body = 0 Also W = = F ⋅ S cosθ = F S cos 90⋅ o^ = 0 Points at a glance ● Consider a projectile of mass m thrown with velocity u making angle with the horizontal. It is projected from the point O and returns to the ground at G. Also M is the highest point attained by it.
(1) In going from O to M , following changes take place – (a) Change in velocity (b) Change in speed (c) Change in momentum (d) Change (loss) in kinetic energy (e) Change (gain) in potential energy (f) Change in the direction of motion (2) On return to the ground, that is in going from O to G , the following changes take place (a) Change in speed = zero (b) Change in velocity = (c) Change in momentum = (d) Change in kinetic energy = zero (e) Change in potential energy = zero (f) Change in the direction of motion = (3) At highest point, the horizontal component of velocity is vx=u cos θ and vertical component of velocity vy is zero. (4) At highest point, linear momentum of a particle; m vx = mu cosθ. (5) Kinetic energy of the particle at the highest point = (6) At highest point, acceleration due to gravity acting vertically downward makes an angle of 90° with the horizontal component of the velocity of the projectile.
(27) Centripetal force is always directed towards the centre of the circular path. (28) When a body rotates with uniform velocity, its different particles have centripetal acceleration directly proportional to the radius. (29) There can be no circular motion without centripetal force. (30) Centripetal force can be mechanical, electrical or magnetic force. (31) Planets go round the earth in circular orbits due to the centripetal force provided by gravitational force of the sun. (32) Gravitational pull of earth provides centripetal force for the orbital motion of the moon and artificial satellites. (33) Centripetal force cannot change the kinetic energy of the body. (34) In uniform circular motion the magnitude of the centripetal acceleration remains constant whereas its direction changes continuously but always directed towards the centre. (35) A pseudo force, that is equal and opposite to the centripetal force is called centrifugal force. (36) The and are directed along the axis of the circular path. Their sense of direction is given by the right hand fist rule as follows : ‘If we catch axis of rotation in right hand fist such that the fingers point in the direction of rotation, then the outstretched thumb gives the direction of and (37) and are called pseudo vectors or axial vectors. (38) For circular motion we have (i) ; (ii) antiparallel to ; (iii) (iv) , (iv) are perpendicular to ; (v) and lie in the same plane
