Projectile motion on an Inclined plane
Projectile motion along an inclined plane: Let us consider an inclined plane that makes an angle 
with the horizontal, as shown in figure.
A particle is projected at time t = 0 with an initial velocity u at an angle 
with the horizontal and it strikes the inclined plane at point A. Let us consider x-axis along the inclined plane and the y-axis perpendicular to it. Then, the component of initial velocity along x and y axis are respectively ux = ucos( – ) and uy = usin( – ) ——– (i)
The component of gravitational acceleration along x and y axis are respectively ax = – gsin and ay = – gcos ———- (ii) [taking downward direction as negative].
If the particle is at point P (x, y) at time t then from equation (i) and (ii) we get,
x = ucos( – )t – 
(gsin)t2 ———- (iii)
and y = usin( – )t – (gcos)t2 ———— (iv)
Time of flight: If T is the time of flight of the projectile above the inclined plane then its displacement along the y-axis is zero in time T. Then from equation (iv) we get,
0 = usin( – )T – (gcos)T2
∴ T = 
.
Range: The range of the projectile on the inclined plane is OA. Then from equation (iii) the range of the projectile on the inclined plane is OA = R = ucos( – )T – (gsin)T2
R = ucos( – )[] – gsin[]2
R = 
[coscos( – ) – sinsin( – )]
R = 
.
Example: 1. A particle is projected from the bottom of an inclined plane of inclination 
with the horizontal. If it strikes the inclined horizontally, then calculate the angle of projection with horizontal.
Let us consider the angle of projection of the particle with horizontal is . Since the velocity of the particle is along the horizontal at the incident point P, hence P is the highest point of the parabolic trajectory, then
PQ = H = 
and OQ = 
= [
]
∴ tan = 
= 
= tan
Or, tan = 2tan
The angle of projection = tan-1(2tan).
2. A particle is projected horizontally with speed u from a point of an incline plane. The plane is inclined at an angle 
with horizontal. How far and when the particle strikes the plane?
Let us consider x and y axis from the point of projection A. The particle strikes the inclined plane at point B (x, y) after time t.
The motion of the particles alone x axis:
Initial speed = u, acceleration = gcos900 = 0
Displacement along x axis is x = ut —– (i)
Motion of the particle along y axis: Initial speed = ucos900 = 0, acceleration = g
Displacement along y axis is y = 
—– (ii)
From diagram 
= tan —– (iii)
From equation (i) and (ii) = 
——— (iv)
From equation (iii) and (iv) we get, tan =
∴ t = 
Therefore the distance travelled by the particle along the incline plane is AB = 
= 
[using equation (iii)]
= xsec
= utsec
= 
. [Putting the value of t]
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