PHYSICS IX-CLASS-CDF
PHYSICS
INCLINED PROJECTILE (14-08-2026)
A body is projected up the inclined plane from the point O with an initial velocity v0 at an angle with
horizontal.
v0 x
x
A
O B
x'
y'
x
y
g sin
g cos
O g B
x1
y1
1. Acceleration along x axis , a x = -gsin α
2. Acceleration along y axis , a y = - g c o s α
3. Component of velocity along x axis
u x = v0cos θ - α
4. Component of velocity along y axis
u y = v0sin θ -α
2v0sin θ - α
5. Time of flight T =
gcosα
6. Range of projectile (OA)
v 20 2v02sin θ - α cosθ
R= sin 2θ - α - sinα . (or) R =
gcos2α gcos 2 α
Narayana CO Schools IIT CO SPARK PROGRAMME 1
, IX-CLASS-CDF PHYSICS
7. For maximum range 2
2
v 20 1- sinα
R max =
gcos 2 α
Down the plane : Here, x and y-directions are down the plane and perpendicular to plane respectively
y
u x
g sin
g g cos
O
u x u cos( ), ax g sin
u y u sin( ), a y g cos
2u sin( )
Proceeding in the similar manner , we get the following results T ,
g cos
8. Suppose two bodies A and B are projected simultaneously from the same point with initial velocities u1
and u2 at angles 1 and 2 with horizontal.
The instantaneous positions of the two bodies are given by
1 2
Body A : x1 u1 cos 1t , y1 u1 sin 1t gt
2
1 2
Body B : x2 u2 cos 2t , y2 u2 sin 2t gt
2
x u1 cos 1 u2 cos 2 t
y u1 sin 1 u2 sin 2 t
y u1 sin 1 u2 sin 2
slope = x u cos u cos
1 1 2 2
y
9. If u1 sin 1 u 2 sin 2 ( initial vertical components) then slope 0
x
y
x
The path is a horizontal straight line
2 IIT CO SPARK PROGRAMME Narayana CO Schools
PHYSICS
INCLINED PROJECTILE (14-08-2026)
A body is projected up the inclined plane from the point O with an initial velocity v0 at an angle with
horizontal.
v0 x
x
A
O B
x'
y'
x
y
g sin
g cos
O g B
x1
y1
1. Acceleration along x axis , a x = -gsin α
2. Acceleration along y axis , a y = - g c o s α
3. Component of velocity along x axis
u x = v0cos θ - α
4. Component of velocity along y axis
u y = v0sin θ -α
2v0sin θ - α
5. Time of flight T =
gcosα
6. Range of projectile (OA)
v 20 2v02sin θ - α cosθ
R= sin 2θ - α - sinα . (or) R =
gcos2α gcos 2 α
Narayana CO Schools IIT CO SPARK PROGRAMME 1
, IX-CLASS-CDF PHYSICS
7. For maximum range 2
2
v 20 1- sinα
R max =
gcos 2 α
Down the plane : Here, x and y-directions are down the plane and perpendicular to plane respectively
y
u x
g sin
g g cos
O
u x u cos( ), ax g sin
u y u sin( ), a y g cos
2u sin( )
Proceeding in the similar manner , we get the following results T ,
g cos
8. Suppose two bodies A and B are projected simultaneously from the same point with initial velocities u1
and u2 at angles 1 and 2 with horizontal.
The instantaneous positions of the two bodies are given by
1 2
Body A : x1 u1 cos 1t , y1 u1 sin 1t gt
2
1 2
Body B : x2 u2 cos 2t , y2 u2 sin 2t gt
2
x u1 cos 1 u2 cos 2 t
y u1 sin 1 u2 sin 2 t
y u1 sin 1 u2 sin 2
slope = x u cos u cos
1 1 2 2
y
9. If u1 sin 1 u 2 sin 2 ( initial vertical components) then slope 0
x
y
x
The path is a horizontal straight line
2 IIT CO SPARK PROGRAMME Narayana CO Schools
