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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