How can the distance traveled during the first 10 second for the graph provided be found? The graph can be found at: http://www.answers.com/topic/velocity-time-graph
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The velocity of an object is the change in displacement per unit time. The instantaneous velocity is the change in displacement divided by the duration of time taken with the duration of time tending to zero. If the displacement can be expressed as a function of time, velocity is the derivative of the function of displacement.
In the website you have provided we are given the velocity as a function of time. It is not possible to find the distance travelled without the exact function that the graph represents. Once you have the function that the graph represents, it can be integrated with respect to time between the intervals of time t = 0 and t = 10 to find the distance travelled in the first 10 seconds.
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