Page 91 - Class-11-Physics-Part-1_Neat

MOTION IN A PLANE 77

This relative velocity vector as shown in
Fig. 4.16 makes an angle θ with the vertical. It is
given by
v 12
tan θ = b = = 0.343
v 35
r
Or,
θ ≅ 19
Therefore, the woman should hold her
umbrella at an angle of about 19° with the
vertical towards the west.
Note carefully the difference between this
Example and the Example 4.1. In Example 4.1,
Fig 4.17 Motion of an object projected with velocity
the boy experiences the resultant (vector
v at angle θ .
sum) of two velocities while in this example, o 0
the woman experiences the velocity of rain If we take the initial position to be the origin of
relative to the bicycle (the vector difference the reference frame as shown in Fig. 4.17, we
of the two velocities). t have :
x = 0, y = 0
4.10 PROJECTILE MOTION o o
Then, Eq.(4.34b) becomes :
As an application of the ideas developed in the
previous sections, we consider the motion of a x = v t = (v cos θ ) t
ox
o
o
projectile. An object that is in flight after being and y = (v sin θ ) t – ( ½ )g t 2 (4.38)
o o
thrown or projected is called a projectile. Such
The components of velocity at time t can be
a projectile might be a football, a cricket ball, a
obtained using Eq.(4.33b) :
baseball or any other object. The motion of a
projectile may be thought of as the result of two v = v = v cos θ o
o
x
ox
separate, simultaneously occurring components
v = v sin θ – g t (4.39)
y
of motions. One component is along a horizontal o o
Equation (4.38) gives the x-, and y-coordinates
direction without any acceleration and the other of the position of a projectile at time t in terms of
along the vertical direction with constant
two parameters — initial speed v and projection
acceleration due to the force of gravity. It was angle θ . Notice that the choice of mutually
o
Galileo who first stated this independency of the o
perpendicular x-, and y-directions for the
horizontal and the vertical components of analysis of the projectile motion has resulted in
projectile motion in his Dialogue on the great a simplification. One of the components of
world systems (1632).
velocity, i.e. x-component remains constant
In our discussion, we shall assume that the throughout the motion and only the
air resistance has negligible effect on the motion
y- component changes, like an object in free fall
of the projectile. Suppose that the projectile is
in vertical direction. This is shown graphically
launched with velocity v that makes an angle
o at few instants in Fig. 4.18. Note that at the point
θ with the x-axis as shown in Fig. 4.17.
o of maximum height, v = 0 and therefore,
y
After the object has been projected, the θ = tan −1 v y = 0
acceleration acting on it is that due to gravity v x
which is directed vertically downward:
Equation of path of a projectile
a = −g j
What is the shape of the path followed by the
Or, a = 0, a = – g (4.36)
x y projectile? This can be seen by eliminating the
The components of initial velocity v are : time between the expressions for x and y as
o
v = v cos θ o given in Eq. (4.38). We obtain:
ox
o
v = v sin θ (4.37)
oy o o
2018-19