Website owner: James Miller
Uniformly accelerated motion
Suppose a body starts from rest and moves in a straight line with a velocity that increases at a constant rate of acceleration, a. The distance, s, that it has traveled after time, t, and its velocity, v, at time, t, are given by the following equations:
v = at
s = ½ at2
Its velocity, in terms of distance traveled, is given by
v2 = 2as
Suppose a body has an initial velocity, v0, and moves in a straight line with a velocity that increases at a constant rate of acceleration, a. The distance, s, that it has traveled after time, t, and its velocity, v, at time, t, are given by the following equations:
v = v0 + at
s = v0t + ½ at2
Its velocity, in terms of distance traveled, is given by
v2 = v02 + 2as
Derivation of formulas. It is obvious that the velocity of the body at time t is given by
1) v = v0 + at
where the acceleration a is the increase in velocity per second and is a constant.
The average velocity of the body from time t = 0 to t = t is given by
The distance s that the body travels in time t is
, or,
Substituting 1) into 3) we obtain
which is the second formula: s = v0t + ½ at2 .
Rearranging 1), we get
Substituting 4) into 3) gives
or
v2 = v02 + 2as
Balls rolling down inclined planes. Suppose a ball rolls down an inclined plane that makes an angle θ with the horizontal. If friction is neglected, the ball will roll with an acceleration, a, given by
a = g sin θ
where g is the acceleration of gravity. Assuming that the ball starts out from rest, the distance, s, that it has rolled after time, t, and its velocity, v, at time, t, are given by
v = at
s = ½ at2
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