Physics Help
Velocity
Velocity
Velocity is a
vector measurement of the rate and direction of motion. The
scalar absolute value (magnitude)
of velocity is
speed. Velocity can also be defined as rate of change of
displacement.
In both
mechanics the average speed v of an object moving a distance d
during a time interval t is described by the simple formula:
-
v = d/t.
The instantaneous velocity vector v of an
object whose position at time t is given by x(t)
can be computed as the
derivative
-
v = dx/dt.
Acceleration is the change of an object's velocity over time. The average
acceleration of a of an object whose speed changes from vi
to vf during a time interval t is given
by:
-
a = (vf - vi)/t.
The instantaneous acceleration vector a of
an object whose position at time t is given by x(t)
is
-
a = d2x/(dt)2
The final velocity vf of an
object which starts with velocity vi and then
accelerates at constant acceleration a for a period of time t
is:
-
vf = vi
+ at
The average velocity of an object undergoing constant
acceleration is (vf + vi)/2.
To find the displacement d of such an accelerating object during a time
interval t, substitute this expression into the first formula to get:
-
d = t(vf +
vi)/2
When only the object's initial velocity is known, the
expression
-
d = vit + (a't2)/2
can be used. These basic equations for final velocity and
displacement can be combined to form an equation that is independent of time:
-
vf2 = vi2
+ 2ad
The above equations are valid for both
classical mechanics and
special relativity. Where
classical mechanics and
special relativity differ is in how different observers would describe the
same situation. In particular, in
classical mechanics, all observers agree on the value of 't' and the
transformation rules for position create a situation in which all
non-accelerating observers would describe the acceleration of an object with the
same values. Neither is true for
special relativity.
The
kinetic energy (movement
energy) of a moving object is linear with both its
mass
and the square of its velocity:
-
The kinetic energy is a
scalar quantity.
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