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Coordinates in horizontal (x) and vertical (y) directions

This approach retains the coordinates used in the normal case (in which projectile returns to the same horizontal level). In this consideration, the description of projectile motion is same as normal case except that motion is aborted in the mid-air by the incline. Had incline been not there, the projectile would have continued with its motion as shown in the figure.

Projectile motion up an incline

Projetile motion up is curtailed by incline.

When the projectile is allowed to return to the projection level, then the point of return is (OQ,0), where OQ is the horizontal range. This position of point of return changes to a new point (x,y), specified by the angle of elevation “α” of the wedge with respect to horizontal as shown in the figure.

Projectile motion up an incline

Projetile motion descrbed with x-axis in horizontal direction and y-axis in vertical direction.

From the triangle OPQ,

cos α = x O P = x R

The range of the projectile is given by :

R = x cos α

The strategy here is to determine “x” i.e. “OQ” considering the motion as normal projectile motion. Thus, we shall first determine “x” and then using above relation, we obtain the relation for the range of flight along the incline. Now, considering motion in horizontal direction , we have :

x = u cos θ X T

where “T” is the time of flight of projectile motion on the incline. It is given as determined earlier :

T = 2 u sin θ α g cos α

Substituting in the epression of “x”, we have :

x = u cos θ X 2 u sin θ α g cos α

x = u 2 2 sin θ α cos θ g cos α

We simplify this relation, using trigonometric relation as given here :

sin C sin D = 2 sin C D 2 cos C + D 2

Comparing right hand side of the equation with the expression in the numerator of the equation of “x”, we have :

C D = 2 θ 2 α

C + D = 2 θ

Adding, we have :

C = 2 θ α

D = α

Thus, we can write :

2 sin θ α cos θ = sin 2 θ α sin α

Substituting the expression in the equation of “x”,

x = u 2 g cos α { sin 2 θ α sin α }

Using the relation connecting horizontal range “x” with the range on incline, “R”, we have :

R = x cos α

R = u 2 g cos 2 α { sin 2 θ α sin α }

Thus, we get the same expression for range as expected. Though the final expressions are same, but the understanding of two approaches is important as they have best fit application in specific situations.

Projection down the incline

A typical projection down the incline is shown with a reference system in which “x” and “y” axes are directions along incline and perpendicular to incline. The most important aspect of the analysis of this category of projectile motion is the emphasis that we put on the convention for measuring angles.

Projectile motion down an incline

Projetile motion thrown from a higher point.

The angle of projection and angle of incline both are measured from a horizontal line. The expression for the time of flight is obtained by analyzing motion in vertical directions. Here we present the final results without working them out as the final forms of expressions are suggestive.

1: Components of initial velocity

u x = u cos θ + α

u y = u sin θ + α

2: Components of acceleration

Components of acceleration due to gravity

Accelration due to gravity forms an angle with y-direction, which is equal to angle of incline.

Questions & Answers

How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
Abdhool Reply
show that 1w= 10^7ergs^-1
Lawrence Reply
what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
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water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
Atinuke Reply

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