# Imagine a particle moving along a straight-line path in...

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Page 1 of 7 Math 1431 - Section 3.1 B – Velocity Imagine a particle moving along a straight-line path in some way. On this line, choose a point of reference, a positive direction and a negative direction. For example, we can choose the line to be the x axis, reference point can be the origin, moving to the right is the positive direction, and left is the negative direction. Let 𝑡𝑡 be a variable representing the time elapsed since some reference time (for example, 𝑡𝑡 = 0 ). Let 𝑠𝑠 ( 𝑡𝑡 ) be the position of the particle at time 𝑡𝑡 measured relative to some reference point (where = 0 ). If the position function 𝑠𝑠 ( 𝑡𝑡 ) is differentiable, then the derivative 𝑠𝑠 ( 𝑡𝑡 ) gives the rate of change or velocity of the position function at time 𝑡𝑡 In symbols, 𝑣𝑣 ( 𝑡𝑡 ) = 𝑠𝑠 ( 𝑡𝑡 ) = 𝑑𝑑𝑠𝑠 𝑑𝑑𝑡𝑡 Speed is the absolute value of velocity. Average velocity over a time interval 𝑎𝑎 𝑡𝑡 𝑏𝑏 is the change in position divided by the change in time: 𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑠𝑠 ( 𝑏𝑏 ) − 𝑠𝑠 ( 𝑎𝑎 ) 𝑏𝑏 − 𝑎𝑎 If the velocity is positive, the object is moving right (positive direction). .