Kinematics

Mechanics is the study of motion. It is divided into two parts kinematics and dynamics. The former is an introduction for the latter. Dynamics contains the laws of motion as the newton's law, that we use to predict the motion of an object from available information about this object. In this lecture we discuss kinematics of a motion's object in one dimension then two dimensions.

1. Position vector and displacement

We simplify the motion of an object by treating this object as a particle. This particle is idealized entity with no size or internal structure. For example treating the sun and the planets as particles is a valid approximation since the sun and the planets radii are much smaller than the distances between them.

To describe the motion of an object, the first step is to establish a coordinate frame or aframe of reference. For motion along a straight line, we select an origin at some point on the line and a positive direction. Measurements are then made relative to this frame of reference. For an object moving on straight line, the position vector is r = x i, where i is the unit vector, x is the coordinate of the moving object.

The displacement Δ r is the change in position. It is the difference between the final position vector r_fand the initial position vector r_i Δ r = r_f - r_i = Δx i.

2. Velocity and speed

The velocity of an object tells us how rapidly the object is moving and what direction it has relative to a reference frame.

2.1. Average velocity
It is defined as v = (r_f - r_i)/(t_f - t_i) =
(Δr)/(Δt)
Where r_f and r_i are the position vectors that locate the object at the time t_f and t_i.
In one dimension v = (Δx)i/(Δt)

2.2. Instantaneous velocity
It is defined as v = lim (r_f - r_i)/(t_f - t_i)
when t_f tends to t_i or

v = lim (Δr)/(Δt)
Δt tends to 0.

If we represent the x(t) curve, we remark that the velocity component is equal to the slope of the line tangent to the curve x(t).



The slope of a line tangent to a curve at a point t_i is the derivative of the curve x(t) with respect to the variable t_i.

v = lim (Δr)/(Δt) = v_x = dx/dt
Δt tends to 0.

dx/dt is called the slope of the graph x versus t at a particular time, and v is the instantaneous velocity at this time.

The velocity is the limiting velocity of the average velocity as the time interval approaches zero. v = lim v = lim Δr)/(Δt dr/dt when δt tends to zero

Velocity is vector quantity and speed is the magnitude of velocity = |v| = |dr/dt|. The SI unit is meter/sec (m/s). Notice velocities can be shown negative, but never speeds.

Examples:
earth surface spins at 4.6 x 10 ³ m/s
center of earth orbits the sun at 3.0 x 10 ³ m/s
human average speed: 1 m/s.

3. Acceleration

The acceleration of an object charactrizes how rapidly its velocity is changing, both im magnitude an direction. It is the rate of change of velocity. To define acceleration, we use average acceleration:
a = (v_f -v_i)/(t_f - t_i) = (Δv)/(Δt)
Where v_f and v_i are the velocity vectors at the time t_f and t_i.

In one dimension a = (Δv_x)i/(Δt)

The Instantaneous acceleration is defined as
a = lim (v_f -v_i)/(t_f - t_i)
when t_f tends to t_i or

a = lim (Δv)/(Δt)
Δt tends to 0.

= a_x = dv_x/dt = d²x/dt²

acceleration is vector quantity. The SI unit is meter/second/second (m/s²). Notice velocities can be shown negative, but its magnitude.

4. Motion with constante acceleration

a = dv/dt    (1)
integrating yields
v = at + v_o     (2)
dx/dt = at + v₀
Another integral gives
x = x_o + v₀t + (1/2) at²    (3)

Eliminating t from (2) and (3) gives:
v² - v_o² = 2a (x - x₀)

To recap: