Page 97 - Class-11-Physics-Part-1_Neat
P. 97
MOTION IN A PLANE 83
as ∆t tends to zero :
∆r d r
v = lim = . It can be written in unit vector notation as :
∆t → 0 ∆t t d
d x d y d z
v = v x i + v y j + v z k where v = t d , v = t d v , z = t d
x
y
When position of an object is plotted on a coordinate system, v is always tangent to
the curve representing the path of the object.
14. If the velocity of an object changes from v to v′in time ∆t, then its average acceleration
v − v' ∆ v
a = =
is given by:
∆t ∆t
The acceleration a at any time t is the limiting value of a as ∆t 0 :
lim ∆ v d v
a = =
∆t → 0 ∆t t d
In component form, we have : a = a x i + a y j + a z k
dv x dv y dv z
where, a x = , a = , a =
z
y
dt dt dt
2
15. If an object is moving in a plane with constant acceleration a = a = a + a 2 and
x y
its position vector at time t = 0 is r , then at any other time t, it will be at a point given
o
by:
1
r = r o + v o t + t a 2
2
and its velocity is given by :
v = v + a t
o
where v is the velocity at time t = 0
o
In component form :
x = x + v t + 1 a t 2
o ox x
2
1
y = y + v t + a t 2
oy
o
2 y
v = v + a t
x ox x
v = v + a t
y oy y
Motion in a plane can be treated as superposition of two separate simultaneous one-
dimensional motions along two perpendicular directions
16. An object that is in flight after being projected is called a projectile. If an object is
projected with initial velocity v making an angle θ with x-axis and if we assume its
o o
initial position to coincide with the origin of the coordinate system, then the position
and velocity of the projectile at time t are given by :
x = (v cos θ ) t
o o
y = (v sin θ ) t − (1/2) g t 2
o o
v = v = v cos θ
x ox o o
v = v sin θ − g t
y o o
The path of a projectile is parabolic and is given by :
y = (tanθ 0 ) x – gx 2
2 v cosθ o ) 2
( o
The maximum height that a projectile attains is :
2018-19
