Question

prove that y = x tan theta [ 1-x/R where (X,y) are coordinates and r is range ​

Answer 1

Answer:

Suppose initial velocity of projection is u and this velocity makes an angle ( theta) with horizontal. ( theta) is called angle of projection.

ux= u cos (theta).

uy= u sin (theta) are x and y components of initial velocity .

We treat x and y motions separately .

x- motion: if we take origin of Cartesian coordinate system at the point of projection and if projectile moves in (x,y) plane , then

at time t, x coordinate of projectile will be ( in the absence of acceleration in horizontal x direction )

x= ux t…………..(1)

y- motion: In this motion there is acceleration , g due to gravity ,in the negative y direction. Therefore , the kinematic equation for this motion will be

y = uy t - (1/2) g t^2……………..(2)

In order to get equation of trajectory we have to eliminate t from the above equations to obtain relation between x and y .

From equation (1), t= x/ux.

Putting this value of t in equation (2), we get

y= (uy/ux)x - (1/2) g (x/ux)^2………….(3) OR

y= x (uy/ux) [1 - (1/2)(g x^2/ux^2)/[(uy/ux)x]] OR

y= x (uy/ux) [ 1- (1/2)g x/ ux uy ]………..(4).

Now, total horizontal distance covered by the projectile is called it's range, R. This distance is covered in time of flight duration .

To obtain time of flight , consider again y motion.

vy= uy - gt.

When vy = 0, then say, time is T/2. Then,

T/2= uy/g OR

Time of flight, T= 2uy/g.

Then, R= ux 2 uy /g………..(5)

Using equation (5) in equation (4), we get

y= x tan (theta) [ 1 - x/R ]……………(6). We have used uy/ux= sin(theta)/cos(theta)= tan (theta)

Explanation:

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