mthsc206-fall-2010-notes-14.4

mthsc206-fall-2010-notes-14.4 - 14.4 Motion in Space...

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Unformatted text preview: 14.4 Motion in Space: Velocity and Acceleration Definition 14.4.1 If the vector-valued function r ( t ) gives the position in space of an object in motion, then v ( t ) = r ( t ) = lim h → r ( t + h )- r ( t ) h is the velocity. The speed of the object is given by v ( t ) = || v ( t ) || = || r ( t ) || . The acceleration a ( t ) is given by a ( t ) = v ( t ) = r ( t ) . Example 14.4.1 The position vector of a point P is given by r ( t ) = 2 t i + 3 t 2 j + t 3 k for ≤ t ≤ 2 . Find the velocity, speed and acceleration at time t . Example 14.4.2 A particle starts at the origin with initial velocity i- j + 3 k . Its acceleration is a ( t ) = 6 t i + 12 t 2 j- 6 t k . Find its position function. Newton’s Second Law of Motion: F = m a . General Projectile Problem: A projectile is fired, with initial velocity v , from a point h feet above ground. If the only force acting on the projectile is caused by the gravitational acceleration g , determine the position function for the projectile after...
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