7. Show that the trajectory of an object thrown at certain angle with the horizontal is a parabola,
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Answer:
Two dimensional motion :
→ In two dimensional motion
r
=x
i
^
+y
j
^
→Velocity
v
=
dt
dx
i
^
+
dt
dy
j
^
⇒
V
=V
x
i
^
+V
y
j
^
→O can be resolved into components.
→U
x
=usinθ
U
y
=usinθ
→ After some time
′
t
′
→V
x
=U
x
+a
x
⋅t=ucosθ+cost
⇒V
x
=ucosθ
→V
y
=U
y
+a
y
t
=ucosθ−gt
→
V
=V
x
i
^
+V
y
j
^
→ After some from, displacement
S=ut+
2
1
at
2
X=ucosθt as a
x
=0
Y=usinθt−
2
1
gt
2
→ As X=ucosθ⋅t
t=
ucosθ
x
→ As Y=usinθt−
2
1
gt
2
=usinθ
ucosθ
x
−
2
1
g
u
2
cos
2
θ
x
2
=tanθ×−
2u
2
cos
2
θ
gx
2
⇒Y=Ax−Bx
2
→ where A=tanθ and
B=
2u
2
cos
2
θ
g
and
both are constants
⇒ The above equation represents a parabola.
→ So path of a projectible is parabola.
→ Velocity along X−direction ucosθ always remains constant as there is no gravity in that direction.
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