Aball is thrown with an initial speed vi at an angle i with the horizontal. the horizontal range of the ball is r, and the ball reaches a maximum height r/6. in terms of r and g, find (a) the time the ball is in motion, (b) the ball’s speed at the peak of its path, (c) the initial vertical component of its velocity, (d) its initial speed, and (e) the angle i . (f) suppose the ball is thrown at the same initial speed found in part (d) but at the angle appropriate for reaching the maximum height. find this height. (g) sup- pose the ball is thrown at the same initial speed but at the angle necessary for maximum range. find this range.

Part a)

Part b)

Part c)

Part d)

Part e)

Part f)

Part g)

Explanation:

Initial speed of the launch is given as

initial speed =

angle = degree

Now the two components of the velocity

similarly we have

Part a)

Now we know that horizontal range is given as

maximum height is given as

so we have

time of flight is given as

Part b)

Now the speed of the ball in x direction is always constant

so at the peak of its path the speed of the ball is given as

Part c)

Initial vertical velocity is given as

Part d)

Initial speed is given as

so we will have

Part e)

Angle of projection is given as

Part f)

If we throw at same speed so that it reach maximum height

then the height will be given as

Part g)

For maximum range the angle should be 45 degree

so maximum range is

the working equation for the angular size is the product of the physical size and 360 degrees divided by the product of two pi and distance from earth. the minimum size would be 0.559 degrees while the maximum size would be 0.49 degrees. 

gas and liquid