Best Canon :D

In order to make a perfect canon which has the greatest horizontal range covered, one needs to keep in mind the angle of the barrel which must be 45 degrees above the horizontal. This angle would be perfect in order to make the canon go as further as it could. We can use the following formula in order to determine the angle at which the canon covers the maximum range:

R = V²sin(2θ)/g

Using this formula, we can determine the value of θ that will maximize the value of sin(2θ), which will maximize the range of the projectile since v and g do not change. The maximum value of sin(2θ) is 1, which occurs at sin(90). Therefore the value of θ must be 45 degrees above the horizontal. 

Newton's Problems

1: Equilibriums
Assumption:
-Positive axes
-∑F = 0 ( ∑Fx = 0 , ∑Fy = 0), a = 0
- no friction
- no air resistance
- rope is weightless

When calculating, divide all the force to x-axis and y-axis,
Fx = max        Fy = may
Fx = 0              Fy = 0

2: Inclines
a) object is at rest
Assumptions:
- postive is along the incline direction
- ∑F = 0, a = 0
- no air resistance
- FN is perpendicular to incline
                     ∑F = ma 
∑Fx = max                ∑F = may
∑Fx = 0                       ∑F = 0
Fgx - f = 0                 FN - Fgy = 0

b) object is moving
Assupmtions:
- positive is along the incline direction
- ∑Fy = 0 , ∑F is not equal to 0
- no air resistance
- FN is perpendicular to incline
use the same method of calculation as when object is at rest

3: Pulley
Assumptions:
- polley is frictionless
- rope is frictionless and weightless
- no air resistance
- 2 systems (2 FBDs)
- T1 = T2
- a of the system is the same (ay1 = ay2)

Note that ∑Fx = 0 for both objects
Do the calculation by dividing the foerces into x and y.
4: Trains
Assumptions:
- Use 3FBDs  to find T
- no air resistance
- ay = 0
- cables are weightless
- positive is in the direction of a
- a is consistant

Assume the whole system as an object to calculate the acceleration
Then use 3 FBDs to find tension amoung the masses

Projectile Motion

Projectile motion is a special case of two dimensional motion with constant acceleration. Here, force due to gravity moderates linear motion of an object thrown at certain angle to the vertical direction. The resulting acceleration is a constant, which is always directed in vertically downward direction.

There is a very useful aspect of two dimensional motion that can be used with great effect. Two dimensional motion can be resolved in to two linear motions in two mutually perpendicular directions : (i) one along horizontal direction and (ii) the other along vertical direction. The linear motion in each direction can, then, be analyzed with the help of equivalent scalar system, in which signs of the attributes of the motion represent direction. The x-axis in the projectile motion represents the horizontal direction of the motion and the y-axis represents the vertical axis.

Flight of base ball, golf ball etc. are examples of projectile motion. In these cases, the projectile is projected with certain force at certain angle to vertical direction. The force that initiates motion is a contact force.