Physics 11th Motion in a Plane

MOTION IN A PLANE

1. Introduction

• In previous chapter we have learned about the motion of any particle along a straight line
• Straight line motion or rectilinear motion is motion in one dimension.Now in this chapter ,we will consider both motion in two dimension and three dimension.
• In two dimensional motion path of the particle is constrained to lie in a fixed plane.Example of such motion motion are projectile shot from a gun ,motion of moon around the earth,circular motion and many more.
• To solve problems of motion in a plane,we need to generalize kinematic language of previous chapter to a more general using vector notations in two and three dimensions.

2.Average velocity

• Consider a particle moving along a curved path in x-y plane shown below in the figue
• Suppose at any time,particle is at the point P and after some time 't' is at point Q where points P and Q represents the position of particle at two different points.
  
  
  
• Position of particle at point P is described by the Position vector r from origin O to P given by
  r=xi+yj
  where x and y are components of r along x and y axis
• As particle moves from P to Q,its displacement would be would be Δr which is equal to the difference in position vectors r and r'.Thus
  Δr = r'-r = (x'i+y'j)-(xi+yj) = (x'-x)i+(y'-y)j = Δxi+Δyj                                          (1)
  where Δx=(x'-x) and Δy=(y'-y)
• If Δt is the time interval during which the particle moves from point P to Q along the curved path then average velocity(v_avg) of particle is the ratio of displacement and corresponding time interval
  
  
  since v_avg=Δr/Δt , the direction of average velocity is same as that of Δr 
• Magnitude of Δr is always the straight line distance from P to Q regardless of any shape of actual path taken by the particle.
• Hence average velocity of particle from point P to Q in time interval Δt would be same for any path taken by the particle.

3.Instantaneous velocity

• We already know that instantaneous velocity is the velocity of the particle at any instant of time or at any point of its path.
• If we bring point Q more and more closer to point P and then calculate average velocity over such a short displacement and time interval then
  
  
  where v is known as the instantaneous velocity of the particle.
• Thus, instantaneous velocity is the limiting value of average velocity as the time interval aproaches zero.
• As the point Q aproaches P, direction of vector Δr changes and aproaches to the direction of the tangent to the path at point P. So instantaneous vector at any point is tangent to the path at that point.
• Figure below shows the direction of instantaneous velocity at point P.
  
  
  
• Thus, direction of instantaneous velocity v at any point is always tangent to the path of particle at that point.
• Like average velocity we can also express instantaneous velocity in component form
  
  
  where v_x and v_y are x and y components of instantaneous velocity.
• Magnitude of instantaneous velocity is
  |v|=√[(v_x)²+(v_y)²]
  and angle θ which velocity vector makes with x-axis is
  tanθ=v_x/v_y
• Expression for instantaneous velocity is
  
  
  Thus, if expression for the co-ordinates x and y are known as function of time then we can use equations derived above to find x and y components of velocity.

4. Average and instantaneous acceleration

• Suppose a particle moves from point P to point Q in x-y plane as shown below in the figure
  
  
  
• Suppose v₁ is the velocity of the particle at point P and v₂ is the velocity of particle at point Q
• Average acceleration is the change in velocity of particle from v₁ to v₂ in time interval Δt as particle moves from point P to Q. Thus average acceleration is
  
  
  Average accelaration is the vector quantity having direction same as that of Δv.
• Again if point Q aproaches point P, then limiting value of average acceleration as time aproaches zero defines instantaneous acceleration or simply the acceleration of particle at that point. Ths, instantaneous acceleration is
  
   
• Figure below shows instantaneous acceleration a at point P.
  
  
  
• Instantaneous acceleration does not have same direction as that of velocity vector instead it must lie on the concave side of the curved surface.
• Thus velocity and acceleration vectors may have any angle between 0 to 180 degree between them.

5. Motion with constant acceleration

• Motion in two dimension with constant acceleration we we know is the motion in which velocity changes at a constant rate i.e, acceleration remains constant throughout the motion
• We should set up the kinematic equation of motion for particle moving with constant acceleration in two dimensions.
• Equation's for position and velocity vector can be found generalizing the equation for position and velocity derived earliar while studying motion in one dimension
  Thus velocity is given by equation
  v=v₀+at                                          (8)
  where
  v is velocity vector
  v₀ is Intial velocity vector
  a is Instantanous acceleration vector
  Similary position is given by the equation
  r-r₀=v₀t+(1/2)at²                                          (9)
  where r₀ is Intial position vector
  i,e
  r₀=x₀i+y₀j
  and average velocity is given by the equation
  v_av=(1/2)(v+v₀)                                          (10)
• Since we have assumed particle to be moving in x-y plane,the x and y components of equation (8) and (9) are
  v_x=v_x0+a_xt                                          (11a)
  x-x₀=v_0xt+(1/2)a_xt²                                          (11b)
  and
  v_y=v_y0+a_yt                                           (12a)
  y-y₀=v_0yt+(1/2)a_yt²                                           (12b)
• from above equation 11 and 12 ,we can see that for particle moving in (x-y) plane although plane of motion can be treated as two seperate and simultanous 1-D motion with constant acceleration
• Similar result also hold true for motion in a three dimension plane (x-y-z)

6. Projectile Motion

• Projectile motion is a special case of motion in two dimension when acceleration of particle is constant in both magnitude and direction
• An object is referred as projectile when it is given an intial velocity which subsequently follows a path determined by gravitational forces acting on it.For example bullet fired from the rifle,a javalin thrown by the athelite etc
• Path followed by a projectile is called its trajectory
• In this section we will study the motion of projectile near the earth surface neglecting the air resistance.
• Acceleration acting on a projectile is constant which is acceleration due to gravity (g=9.81 m/s) directed along vertically downward direction.
• we shall treat the projectile motion ina cartesian co-ordinates system taking y axis in vertically upwards direction and x axis alomg horizontal directions
• Now x and y components of acceleration of projectile is
  a_x=0 and a_y=-g
  Since acceleration in horizontal direction is zero,this shows that horizontal component of velocity is constant and verical motion is simply a case of motion with constant acceleration.
• Suppose at time t=0 object is at origin of co-ordinate system and velocity components v_0x and v_0y.From above components of acceleration are a_x=0 and a_y=-g.From equation 11 and 12 in the previous section components of position and velocity are
  x=v_0xt                                          (14a)
  v_x=v_0x                                           (14b)
  and v_y=v_0y-gt                                          (15b)
  y=v_0y-(1/2)gt²                                          (15a)
• Figure below shows motion of an object projected with velocity v₀ at an angle θ₀
  
  
  
• In terms of initial velocity v₀ and angle θ₀ components of initial velocity are
  v_0x=v₀cosθ₀                                          (16a)
  v_0y=v₀sinθ₀                                           (16b)
• Using these relations in equation 14 and 15 we find
  x=(v₀cosθ₀)t                                          (17a)
  y=(v₀sinθ₀)t-(1/2)gt²                                           (17b)
  v_x=v₀cosθ₀                                           (17c)
  v_y=v₀sinθ₀-gt                                           (17d)
  Above equations describe the position and velocity of projectile as shown in fig 5 at any time t.

(A)Equation of Path of projectile(Trajectory)

• From equation 17a
  t=x/v₀cosθ₀
  now putting this value of t in equation 17b,we find
  y=(tanθ₀ )x-[g/2(v₀cosθ₀)²]x²                                           (18)
  In equation (18),quantities θ₀,g and v₀ are all constants and equation (18) can be compared with the equation
  y=ax-bx²
  where a and b are constants
• This equation y=ax-bx² is the equation of the parabola.From this we conclude that path of the projectile is a parabola as shown in figure 5

B) Time of Maximum height

• At point of maximum height v_y=0.Thus from equation (17d)
  v_y=v₀sinθ₀-gt
  0=v₀sinθ₀-gt -
  or t_m=v₀sinθ₀/g                                          (19)
• Time of flight of projectile which is the total time during which the projectile is in flight can be obtained by putting y=0 because when projectile reaches ground ,verical distance travelled is zero.This from equation (17b)
  
  t_f=2(v₀sinθ₀)/g                                          (20)
  or
  t_f=2t_m
• Maximum height reached by the projectile can be calculated by substituting t=t_m in equation 17b
  y=H_m=(v₀sinθ₀)(v₀sinθ₀/g)-(g/2)(v₀sinθ₀/g)²
  or
  H_m=v₀²sin²θ₀/2g                                          (21)

(C) Horizontal Range of Projectile

• Since acceleration g acting on the projectile is acting vertically ,so it has no component in horizontal direction.
• So, projectile moves in horizontal direction with a constant velocity v₀cosθ₀. So range R is
  R=OA=velocity x time of flight
  
   
• Maximum range is obtained when sin2θ₀=1 or θ₀=45⁰. Thus when θ₀=45⁰ maximum range achieved for a given initial velocity is (v₀)²/g.

7. Uniform circular motion

• When an object moves in a circular path at a constant speed then motion of the object is called uniform circular motion.
• In our every day life ,we came across many examples of circular motion for example cars going round the circular track and many more .Also earth and other planets revolve around the sun in a roughly circular orbits
• Here in this section we will mainly consider the circular motion with constant speed
• if the speed of motion is constant for a particle moving in a circular motion still the particles accelerates becuase of costantly changing direction of the velocity.
• Here in circular motion ,we use angular velocity in place of velocity we used while studying linear motion

(A) Angular velocity

• Consider an object moving in a circle with uniform velocity v as shown below in the figure
  
  
  
  
  The velocity v at any point of the motion is tangential to the circle at that point.Let the particle moves from point A to point Balong the circumference of the circle .The distance along the circumference from A to B is
  s=Rθ                                          (23)
  Where R is the radius of the circle and θ is the angle moved in radian's
• Magnitude of velocity is
  v=ds/dt=Rdθ/dt                                          (24)
  Since radius of the circle remains constant quantity,
  ω=dθ/dt                                          (25)
  is called the angular velocity defined as the rate of change of angle swept by radius with time.
• Angular velolcity is expressed in radians per second (rads^-1)
• From equation 24 and 25,we find the following
  v=ωR                                          (26)
• Thus for a particle moving ain circular motion ,velocity is directly proportional to radius for a given angular velocity
• For uniform circular motion i.e, for motion with constant angular velocity the motion would be periodic which means particle passes through each point of circle at equal intervals of time
• Time period of motion is given by
  T=2π/ω                                          (27)
  Since 2π radians is the angle θ in one revolution
• If angular velocity ω is constant then integrating equation (25) with in limits θ₀ to θ,we find
  
  
  where θ₀ is the angular position at time t₀ and θ is the angular position at time t .The above equation is similar to rectilinear motion result x-x₀=v(t-t₀)

(B) Angular acceleration

• Angular acceleration is defined as the rate of change of angular velocity moving in circular motion with time.
  Thus
  α=dω/dt=d²θ/dt²                                          (29)
  Unit of angular acceleration is rads^-2
• For motion with constant angular acceleration
  
  
  or
  ω=ω₀+α(t-t₀)                                          (30)
  where ω₀ is the angular velocity at time t₀
  Again since
  ω=dθ/dt
  or dθ=ωdt then from equation 30
  
  If in the begining t₀=0 and θ₀=0 the angular position at any time t is given by
  θ=ωt+(1/2)αt²
  This result is of the form similar to what we find in case of uniformly accelerated motion while studying rectilinear motion

8. Motion in three dimensions

• We have already studied physical quantities like displacement ,velocity,acceleration etc in one and two dimension
• In this topic ,we will generalize our previous knowlegde of motion in 1 and 2 -dimension to three dimension's
• As we have used vectors to represent motion in a plane,we can freely use vectors and its properties in 3-dimension as we have done in case of motion in a plane 
• In three dimensions ,we have three units vectors i ,j and k associated with each co-ordinate axis of cartesian co-ordinates system shown below in the figure
  
  
  
• Consider a particle moving in 3-D space .Let P be its position at any point t.Position vector of this particle at point P would be
  r=xi+yj+zk
  Where x,y and z are co-ordinates of point P
• Similarly velocity and acceleration vectors of particle moving in 3-D space are
  v=v_xi+v_yj+v_zk where v_x=dx/dt,v_y=dy/dt and v_z=dz/dt
  and
  a=v_xi+a_yj+a_zk
  where a_x=dv_x/dt,a_y=dv_y/dt and a_z=dv_z/dt
• All the relations we have derived incase of motion in plane are valid for 3-D motion with one added co-ordinate