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When Acceleration is Given in Terms of x

If acceleration is a function ofsowe can find the velocity by integration:

and then find the displacement by integrating again

If the acceleration is a function ofwe cannot do this becauseis an unknown function ofWe would have

We can however use the chain rule to express the acceleration as a function of

Now the equationbecomesWe can separate the variables and integrate.

Example

The acceleration of a particle is given in terms ofbyIf the maximum speed of the particle is 10m/s find an expression forin terms of

When the particle moves to the right it accelerates becauseWhen andis max. We can takeand