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We all know what cinema is. The word cinema comes from the greek work Kinema which means motion or movement, and that is why the cinema is also known as motion picture as it's not a static picture but pictures in motion. The french word cinematique has been changed into kinematics to represent the study of motion. Kinematics does not care about what caused the motion (the forces and all that was main subjects in statics) but how the motion itself works and what are the laws governing this concept in nature.
The main things we deal with here are about position (r), velocity (v), acceleration (a), and time. In simple words, the rate of change in position is speed and if we have direction for speed, it is called velocity; the rate of change in speed is called acceleration. That's it, you got it all! Hey maybe it is not moving from a point to another, maybe it's just spinning around itself! rotation is also motion and is dealt with in kinematics. If a body does not have any rotational motion, we can call it a particle, and we must know that all tiny bits of the body in a particle have the same position, velocity, and acceleration at a given time. If several particles are glued to each other that makes it a rigid body. Now, if a rigid body has rotational movement, then each one of the particles composing it can have a different displacement, velocity, and acceleration. Think of throwing a rolling rigid ball; it is turning and as a whole it has a translation motion too; now think of different parts of this rigid ball, they cannot have the same speed as they are also rotating and the resultant of all their speed in rotation and translation cannot be all the same. This is what makes dynamics both interesting and confusing to me!
If the movements of the particles are happening in a straight line, we are dealing with a linear system or rectilinear system. A particle's position can be shown by the three coordinates in a rectangular system (x,y,z). The position of this article can be shown in the form of a vector as in r = xi + yj + zk. In this form, the first two derivatives of the position vector result in velocity and acceleration vectors. So,
r = xi + yj + zk
v = ẋi + ẏj + żk
a = ẍi + ÿj + z̈k
Acceleration is usually constant in what we are dealing with. An example of a constant acceleration is the gravitation acceleration, g. The acceleration being constant, can be drawn out of the integral from the equations and results in nicer formula for calculating the speed and position of the moving particle. Therefore, the following equations:
The velocity of the particle at any given position is shown by two radial (parallel to er) and transverse components (perpendicular to er). This means we can also have a unit vector in the perpendicular direction to er, and name it eθ:
r = rer
v = vrer + vθeθ = r'er + rθ'eθ
a = arer + aθeθ = (r" - rθ'2)er + (rθ" + 2r'θ')eθ
In a curvilinear motion, at any given time, a particle has instantaneous speed and velocity which is directed tangentially to the path of the particle and they are called tangential velocity (vt) and tangential acceleration (at). Also, the same particle experiences an inward acceleration toward the center of its motion, due to the force that moves it in that curvilinear manner, which is called normal acceleration (an). The same of the two above will give the resultant speed and acceleration. If the curvilinear motion is around a fixed circular path, the motion is called circular motion, rotational motion, or angular motion. In a rotational movement, instead of s, v, and a we have θ (angular position), ω (angular velocity), and α (angular acceleration). Linear motion position, velocity, and acceleration can be calculated as a product of r (distance to the center in angular motion) and those of angular motion, e.g. s = rθ.
a(t) = a0
v(t) = v0 + a0t
s(t) = s0 + v0t + a0t2/2
v2(t) = v02 + 2a0(s - s0)
Now, if you throw a ball, it certainly wouldn't just move in a straight line, the ball moves in a curvilinear motion. But still, this motion can also be shown in a rectilinear system. Sometimes, it may be also convenient to show these physical movements in a polar coordinate too. When we deal with polar system, the position of a particle can be shown by its distance from the origin called radius, and the angle it makes with the horizontal axis θ, or the unit vector er that is in the direction of the line connecting origin to the particle position.
The kinematics equation of this type of motion can be derived from the laws of uniform acceleration and conservation of energy. Here is a nice page (accessed 9/12/2015) that provides information, equations, examples, etc. on this subject.
The other class of motion is the projectile motion which is the motion under constant acceleration (gravity). A general projectile in which a particle is set into motion with a velocity v0 (impact velocity) at an angle θ. If we neglect the air drag, then in such motion, the trajectory of the particle is always parabolic, and the particle position's range until it falls down is maximum if the launch angle θ is at 45 degrees. It is interesting that in such motion, when the particle reaches its highest position, the time it will take for it to hit the ground (in a projectile motion) is the same time as if you drop a stationary object from the same position to hit the ground!
Some notes:
If the question asked for initial position, speed, or acceleration then solve for t = 0
If asked for maximum speed, find derivative of speed function and put it equal to zero, find the t and solve the speed function for that t
It is similar to above when the maximum position, or acceleration is asked for.
If the motion is under constant acceleration, the acceleration function should not have any variable in it.
To convert from km/h to m/s, divide by 3.6
Now, we may proceed to the subject of Kinetics.
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