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Unformatted text preview: John Ellison, UCR p. 1 Notes on Chapter 2 Velocity and Speed The position of an object is denoted by its coordinate x(t) along an axis, on which we have defined an origin (the point defined as x = 0). As the body moves, its coordinate changes and hence x is a function of time t . Note that x can be positive or negative - this is the distinction between a coordinate and a distance ( x is not a distance!). When an object moves from coordinate x 1 to coordinate x 2 we define its displacement Δ x by The average velocity v avg is defined by where Δ t is the time interval that elapses between the times t 1 and t 2 at which the object is at positions x 1 and x 2 . The velocity of an object at a single instant in time is called the instantaneous velocity v (or simply the velocity .) It is defined by i.e. it is just the derivative of x with respect to t , or, in other words, the slope of the tangent to the curve of x vs. t at a particular point on the curve corresponding to time t . Note that the displacement in a time interval Δ t is x ( t = Δ t ) - x ( t ), so we can write the last equation as which you should recognize as the definition of the derivative. Example The plot of x as a function of t below shows the graphical interpretation of (a) the average velocity over the interval t = 1 s to t = 4 s, and (b) the (instantaneous) velocity at t = 4 s. Note that the average velocity is the slope of the line joining the two points between t = 0 and t = 4 s. The velocity at t = 4 s is the slope of the red line, which is tangent to the x(t) curve at t = 4 s. There is a value of velocity associated with every instant of time. In this example the velocity from 0-4 s is always positive which means that x increases with time. A negative velocity means that x decreases with time. The speed s (or instantaneous speed ) is the magnitude of velocity. However, average speed s avg is not the magnitude of the average velocity. It is defined by x = x 2 x 1 v avg = x t = x 2 x 1 t 2 t 1 v = lim t x t = dx dt s avg = total distance t v at t = 4 s = slope of this line = dx dt (b) (a) v = lim t x t t x t t = dx dt John Ellison, UCR p. 2 Acceleration When a particle's velocity changes, it is said to undergo acceleration (or to accelerate). For motion along an axis the average acceleration a avg over a time interval Δ t is defined by The instantaneous acceleration (or simply acceleration ) is the derivative of the velocity with respect to time: Using the definition of velocity, we find i.e. the acceleration is the second derivative of x with respect to time. When the velocity and acceleration are in the same direction (same signs) the particle is speeding up, but if the velocity and acceleration are in opposite directions (opposite signs) the particle is slowing down....
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- Fall '07
- Acceleration, Velocity, m/s, John Ellison