Kinematics:Motion in two dimensions
In motion of one dimension , that is along a straight line, such as free-fall, we defined
three kinematics quantities: position vector, velocity, ans acceleration.
We will use them to describe a motion in two dimensions
such as a projectile. A motion in two dimensions occurs in a plane.
Contrary to motion in one dimension, where velocity acceleration are
along the same line, in two dimension motion, they are not
necessarily along the same line as in the uniform circular motion, where
the velocity is tangential and the acceleration is radial.
1. velocity and acceleration
The position vector r locates an object relative to the origin of
a reference frame. In two dimensions, we have:
r = x i + y j
Position vector in two dimensions:
x and y are the object's coordinates. Notice that the graph used in two dimensions of
y(x), whereas in one dimension it is x(t).
The displacement Δr = rf - ri is directed from
the object's initial position to the final position. In component form:
Δr = Δ(x i + yj) = Δx i + Δy j
1.1. Velocity
An object's average velocity v, for a time interval Δt,
is the rate in change of the displacement:
v = Δr/Δt =
(Δx i + Δy j)/Δt = (Δx/Δt) i + (Δy/Δt) j .
v is parallel to Δr. The direction of the average velocity and
the displacement is the same.
The velocity is defined as the limiting value of the average velocity
as the time interval approaches 0.
v = lim v = lim Δr/Δt = dr/dt
Δt tends to 0
In component form:
v = (dx/dt)i + (dy/dt)j = vx i + vy j
The magnitude of the velocity v is then equal to:
v = [vx2 + vy2]1/2
Vector velocity:
v = (dx/dt)i + (dy/dt)j = vx i + vy j
Magnitude of vector velocity:
v = [vx2 + vy2]1/2
As the time interval Δt approaches zero (tf approaches ti),
the average velocity v approaches the velocity v that becomes along the tangent
line on the path and points in the direction of motion.
The velocity v, at any point on an object's path is directed parallel to a line
tangent to the path and points in the
direction of motion.
The direction in which the object goes is given by:
tan θ = vy/vx. Where θ
is the angle between the x-axis and the vector velocity v.
The angle θ is positive when measured counterclockwise from the
x-axis to v.
1.2. Acceleration
An object's average acceleration a, for a time interval Δt,
is the rate in change of the velocity:
a = Δv/Δt =
(Δvx i + Δvy j)/Δt = (Δvx/Δt) i + (Δvy/Δt) j.
a is parallel to Δv. The direction of the average acceleration and
the change in velocity is the same.
The acceleration is defined as the limiting value of the average acceleration
as the time interval approaches 0.
a = lim a = lim Δv/Δt = dv/dt
Δt tends to 0
In component form:
a = (dvx/dt)i + (dvx/dt)j = ax i + ay j
The magnitude of the acceleration a is then equal to:
a = [ax2 + ay2]1/2
Vector acceleration:
a = (dvx/dt)i + (dvy/dt)j = ax i + ay j
Magnitude of vector acceleration:
a = [ax2 + ay2]1/2
As the time interval Δt approaches zero,
the average acceleration a approaches the acceleration a and points in the direction
given by the limiting value where the velocity is along the tangent to the path .
The acceleration a at any point on an object's path is directed toward
the center of curvature.
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