Question

find an equation for the range (defined as the x displacement of the projectile when it reaches y = 0) for a cannon firing from y = 0. Leave everything as variables, such as v_i for the initial velocity, and θ for the launch angle.

After you are confident in your equation you must be able to predict where the projectile will land on the first try.

1. Write out an equation that gives the horizontal position of the object as a function of time.
2. Is the total time the object spends in flight determined by what’s going on horizontally or what’s going on vertically? Come up with an expression for the total time the object spends in flight (the time it takes the object to return to the level from which it was launched), in terms of v_i and θ.
3. Substitute your expression from 2 into your equation in 1, which should produce what we call the range equation – an equation that tells you how far horizontally the object travels, in terms of v_i and θ. You may find the following relation useful: sin theta, cos theta= 1/2 sin 2theta.

Please show the steps and number it properly. Thank you for answering the question. Will definetly give a thumbs up.  

Answer 1

see the diagram :

initial velocity =

angle of projection =

= horizontal acceleration = 0

= vertical acceleration = -g = - 9.81m/s/s.

initial vertical velocity =

horizontal velocity =

When it comes to same level, then time spent in air = TIme of flight = T

in this time, the vertical displacement is zero.

using kinematic equation for vertical motion :

. [answer]

The hozontal distance travelled in time of flight is R.

using uniform motion for horizontal motion :

the horizontal distance as function of t is

[answer]

The horizontal range R = [answer]