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Home Real and Hyperreal Numbers Slope and Velocity; the Hyperreal Line Average Velocity  
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Average Velocity
The whole process can also be visualized in another way. Let t represent time, and suppose a particle is moving along the yaxis according to the equation y = t^{2}. That is, at each time t the particle is at the point t^{2} on the yaxis. We then ask: what is meant by the velocity of the particle at time t_{0}? Again we have the difficulty that the velocity is different at different times, and the calculus is needed to answer the question in a satisfactory way. Let us consider what happens to the particle between a time t_{0 }and a later time t_{0} + Δt. The time elapsed is Δt, and the distance moved is Δy = 2t_{0} Δt + (Δt)^{2}. If the velocity were constant during the entire interval of time, then it would just be the ratio Δy/Δt. However, the velocity is changing during the time interval. We shall call the ratio Δy/Δt of the distance moved to the time elapsed the "average velocity" for the interval; The average velocity is not the same as the velocity at time t_{0} which we are after. As a matter of fact, for t_{0} > 0, the particle is speeding up; the velocity at time t_{0} will be somewhat less than the average velocity for the interval of time between t_{0} and t_{0} + Δt, and the velocity at time t_{0} + Δt will be somewhat greater than the average. But for a very small increment of time Δt, the velocity will change very little, and the average velocity Δy/Δt will be close to the velocity at time t_{0}. To get the velocity v_{0} at time t_{0}, we neglect the small term Δt in the formula v_{ave} = 2t_{0} + Δt, and we are left with the value v_{0} = 2t_{0}. When we plot y against t, the velocity is the same as the slope of the curve y = t^{2}, and the average velocity is the same as the average slope.


Home Real and Hyperreal Numbers Slope and Velocity; the Hyperreal Line Average Velocity 