motion in two dimension Position of points in two dimensions is represented in vector form rl =x1i+Y1j and F2 = x2t+Y2j If particle moves from rl position to r2 position then Displacement is final position — initial position (Xli+ Ylj ) —Yi)j = (x2 — Xl)i + (Y2 If x2 — Xl = dx and Y2 — Yl = dy s = dxi + dyj Now d' dx dy dt dt dt dx/dt = vx is velocity of object along x-axis and dy/dt = vy is velocity along y-axis 13 = vxt + vyj Thus motion in two dimension is resultant of motion in two independent component motions taking place simultaneously in mutually perpendicular directions. Magnitude of velocity Angle between velocity and x-axis is tano = dt Magnitude of acceleration dt
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Equation of motion in vector form 1 = fit + —ät2 13 = u + at 1 * = FO + 7t + —ät2 r average velocity — 2 v2 — u2 = 2ä• (F —Fo) Relative velocity 0' o x x Two frame of reference A and B as shown in figure moving with uniform velocity with respect to each other. Such frame of references are called inertial frame of reference. Suppose two observers, one in from A and one from B study the motion of particle P. Let the position vectors of particle P at some instant of time with respect to the origin O of frame A be Fpm And that with respect to the origin O' of frame B be rp„B The position vector of O' w.r.t O is From figure it is clear = OPP = 00' OP = 00' + orp = O'P + 00' • = + rB,A
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Differentiating above equation with respect to time we get '(EPA) = Fi(FPB) + (FBA) d VBA = Vp,B + 13B A Here Vp,A is the velocity of the particle w.r.t frame of reference A, Vp,B is the velocity of the particle w.r.t. reference frame B and VB,A is the velocity of frame of reference B with respect to frame A Suppose velocities of two particles A and B are respectively VA and VB relative to frame of reference then velocity (VAB) of A with respect to B is VAB = VA — VB And velocity VBA of B relative to A is VBA Thus And Il'ABl = löBAl VA2 + — 2VAVBCOS0 IVABI = Also angle a made by the relative velocity with VA is given by Vesin9 tana — VA — VBcos9
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Projectile motion A projectile is a particle, which is given an initial velocity, and then moves under the action of its weight alone. When object moves at constant horizontal velocity and constant vertical downward acceleration, such a two dimension motion is called projectile. The projectile motion can be treated as the resultant motion of two independent component motion taking place simultaneously in mutually perpendicular directions. One component is along the horizontal direction without any acceleration and the other along the vertical direction with constant acceleration due to gravitational force. Important terms used in projectile motion When a particle is projected into air, the angle that the direction of projection makes with horizontal plane through the point of projection is called the angle of projection, the path, which the particle describes, is called the trajectory, the distance between the point of projection and the point where the path meets any plane draws through the point of projection is its range, the time that elapses in air is called as time of flight and the maximum distance above the plane during its motion is called as maximum height attained by the projectile Analytical treatment of projectile motion p u 0 o Consider a particle projected with a velocity u of an angle 6 with the horizontal earth's surface. If the earth did not attract a particle to itself, the particle would describe a straight line, on account of attraction of earth, however, the particle describes a curve path Let us take origin at the point of projection and x-axis along the surface of earth and perpendicular to it respectively shown in figure Here gravitational force is the force acting on the object downwards with constant acceleration of g downwards. There if no force along horizontal direction hence acceleration along horizontal direction is zero
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Motion in x —direction Motion in x-direction with uniform velocity At t = 0, Xo = 0 and ucose Position after time t , x = xo + ux t — — — eq(4) Vy2 = 2 2 —2gy — — — —eq(5) IX = (ucosé) t Velocity at t , Vx = ux -eq(l) uCOS9 eq(2) Motion in y-direction: Motion in y-direction is motion with uniform acceleration 'When t = 0, Yo = O, uy = using and ay After time Vy — Vy = using — gt Also, -----eq(3) 1 YO uyt Vy2 = u}, + 2ayt2 u sm 9 Time of flight(T) : Time of flight is the time duration which particle moves from O to O'. From equation (4) 1 — gT2 0 = usin9T — 2usin0 — — — —eq(6)
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Range of projectile (R): Range of projectile distance travelled in time T, i.e. R=VxxT R = ucos9T Since 2sinecose = sin29 Maximum height reached (H): 2usin9 R = ucos0 u2sin20 — — — —eq(7) At the time particle reaches its maximum height velocity of particle becomes parallel to horizontal direction i.e. Vy = O, when Y = H From equation (5) Equation of trajectory: u2sin29 — 2gH u2sin29 2g The path traced by a particle in motion is called trajectory and it can be known by knowing the relation between X and Y From equation (1) and (4) eliminating time t we get — Y = Xtan8 This is trajectory of path and is equation of parabola. So we can say the path of particle is parabolic Velocity and direction of motion after a given time: After time 't' Vy = ucos9 and Vy = using - gt
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Hence resultant velocity vx2 + V = u2cos28 + (usin8 — gt)2 If direction of motion makes an angle a with horizontal Vy tana — using — gt ucose usin9 — gt a = tan- UCOSO Velocity and direction of motion at a given height At height , Vx = ucosé And Resultant velocity u2sin29 — 2gh 142 + 1/2 u2 — 2gh Note that this is the velocity that a particle would have at height h if it is projected vertically from ground with u
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