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| A ] measurement of the rate and direction of motion. The scalar absolute value of velocity is ]. | A ] measurement of the rate and direction of motion. The scalar absolute value of velocity is ]. | ||
| In ] (that is, when the velocities involved are significantly less than the ]) the average speed ''v'' of an object moving a distance ''d'' during a time interval ''t'' is described by the simple formula: | In ] (that is, when the velocities involved are significantly less than the ]) the average speed ''v'' of an object moving a distance ''d'' during a time interval ''t'' is described by the simple formula: | ||
| :''v'' = ''d/t''. | :''v'' = ''d/t''. | ||
| The instantaneous velocity vector '''v''' of an object whose position at time ''t'' is given by '''x'''(''t'') can be computed as the ] | The instantaneous velocity vector '''v''' of an object whose position at time ''t'' is given by '''x'''(''t'') can be computed as the ] | ||
| :'''v''' = d'''x'''/d''t''. | :'''v''' = d'''x'''/d''t''. | ||
| ] is the change of an object's velocity over time. The average acceleration of ''a'' of an object whose speed changes from ''v''<sub>''i''</sub> to ''v''<sub>''f''</sub> during a time interval ''t'' is given by: | ] is the change of an object's velocity over time. The average acceleration of ''a'' of an object whose speed changes from ''v''<sub>''i''</sub> to ''v''<sub>''f''</sub> during a time interval ''t'' is given by: | ||
| :''a'' = (''v''<sub><i>f</i></sub> - ''v''<sub><i>i</i></sub>)/''t''. | :''a'' = (''v''<sub><i>f</i></sub> - ''v''<sub><i>i</i></sub>)/''t''. | ||
| The instantaneous acceleration vector '''a''' of an object whose position at time ''t'' is given by '''x'''(''t'') is | The instantaneous acceleration vector '''a''' of an object whose position at time ''t'' is given by '''x'''(''t'') is | ||
| :'''a''' = d<sup>2</sup>'''x'''/(d''t'')<sup>2</sup> | :'''a''' = d<sup>2</sup>'''x'''/(d''t'')<sup>2</sup> | ||
| The final velocity ''v''<sub>''f''</sub> of an object which starts with velocity ''v''<sub>''i''</sub> and then accelerates at constant acceleration ''a'' for a period of time ''t'' is: | The final velocity ''v''<sub>''f''</sub> of an object which starts with velocity ''v''<sub>''i''</sub> and then accelerates at constant acceleration ''a'' for a period of time ''t'' is: | ||
| :''v''<sub><i>f</i></sub> = ''v''<sub><i>i</i></sub> + ''at'' | :''v''<sub><i>f</i></sub> = ''v''<sub><i>i</i></sub> + ''at'' | ||
| The average velocity of an object undergoing constant acceleration is (''v''<sub><i>f</i></sub> + ''v''<sub><i>i</i></sub>)/2. To find the displacement ''d'' of such an accelerating object during a time interval ''t'', substitute this expression into the first formula to get: | The average velocity of an object undergoing constant acceleration is (''v''<sub><i>f</i></sub> + ''v''<sub><i>i</i></sub>)/2. To find the displacement ''d'' of such an accelerating object during a time interval ''t'', substitute this expression into the first formula to get: | ||
| :''d'' = ''t''(''v''<sub><i>f</i></sub> + ''v''<sub><i>i</i></sub>)/2 | :''d'' = ''t''(''v''<sub><i>f</i></sub> + ''v''<sub><i>i</i></sub>)/2 | ||
| When only the object's initial velocity is known, the expression | When only the object's initial velocity is known, the expression | ||
| :''d'' = ''v''<sub><i>i</i></sub>''t'' + (''a''''t''<sup>2</sup>)/2 | :''d'' = ''v''<sub><i>i</i></sub>''t'' + (''a''''t''<sup>2</sup>)/2 | ||
| can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time: | can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time: | ||
| :''v''<sub><i>f</i></sub><sup>2</sup> = ''v''<sub><i>i</i></sub><sup>2</sup> + 2''ad'' | :''v''<sub><i>f</i></sub><sup>2</sup> = ''v''<sub><i>i</i></sub><sup>2</sup> + 2''ad'' | ||
| ⚫ | These simple equations become more complicated as velocities approach the speed of light, where the effects of ] start to become significant. | ||
| ] | |||
| ⚫ | These simple equations become more complicated as velocities approach the speed of light, where the effects of ] start to become significant. | ||
Revision as of 20:34, 31 January 2002
A vector measurement of the rate and direction of motion. The scalar absolute value of velocity is speed.
In Classical mechanics (that is, when the velocities involved are significantly less than the speed of light) the average speed v of an object moving a distance d during a time interval t is described by the simple formula:
- v = d/t.
The instantaneous velocity vector v of an object whose position at time t is given by x(t) can be computed as the derivative
- v = dx/dt.
Acceleration is the change of an object's velocity over time. The average acceleration of a of an object whose speed changes from vi to vf during a time interval t is given by:
- a = (vf - vi)/t.
The instantaneous acceleration vector a of an object whose position at time t is given by x(t) is
- a = dx/(dt)
The final velocity vf of an object which starts with velocity vi and then accelerates at constant acceleration a for a period of time t is:
- vf = vi + at
The average velocity of an object undergoing constant acceleration is (vf + vi)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get:
- d = t(vf + vi)/2
When only the object's initial velocity is known, the expression
- d = vit + (a't)/2
can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time:
- vf = vi + 2ad
These simple equations become more complicated as velocities approach the speed of light, where the effects of special relativity start to become significant.