Explanation Formula Used: If the trajectory of a particle is given by ( x ( t ) , y ( t ) ) , then its speed is given by v ( t ) = [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 Second derivative test: The point for which f ′ ( t ) = 0 is the critical point for the function and if the second derivative of the function is positive at this point, then f has a local minimum at that point. But, if for that critical point the second derivative is negative, then f has local maximum at that point. Calculations: The trajectory of the particle for t ≥ 0.5 is given by c ( t ) = ( t 3 , t − 2 ) Then, x ( t ) = t 3 y ( t ) = t − 2 So, x ′ ( t ) = 3 t 2 y ′ ( t ) = − 2 t − 3 Now, the speed of the particle is evaluated as v ( t ) = [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 = ( 3 t 2 ) 2 + ( − 2 t − 3 ) 2 = 9 t 4 + 4 t − 6 Now, the square root function is an increasing function. Hence, you can say that the minimum speed of the particle occurs at the value t for which the function f ( t ) = 9 t 4 + 4 t − 6 attains its minimum value. So, we have to first find the critical points for the function f ( t ) = 9 t 4 + 4 t − 6 on t ≥ 0...

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Author: Rogawski

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Chapter 12.2, Problem 22E

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To find: minimum speed of a particle with trajectory

$c (t) = (t^{3}, t^{- 2}) for t \geq 0.5$ .

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Chapter 12.2, Problem 21E

Chapter 12.2, Problem 23E

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minimum speed of a particle with trajectory c ( t ) = ( t 3 , t − 2 ) for t ≥ 0.5 .

Solution Summary: The author explains the formula used to find the minimum speed of a particle with trajectory c (t ) = ( t 3, 2 ). The square root function is an increasing function.

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To find: minimum speed of a particle with trajectory

$c (t) = (t^{3}, t^{- 2}) for t \geq 0.5$ .

Want to see the full answer?

Chapter 12 Solutions

minimum speed of a particle with trajectory c ( t ) = ( t 3 , t − 2 ) for t ≥ 0.5 .

Solution Summary: The author explains the formula used to find the minimum speed of a particle with trajectory c (t ) = ( t 3, 2 ). The square root function is an increasing function.

Videos

To find: minimum speed of a particle with trajectory c(t)=(t3,t−2) for t≥0.5.

Want to see the full answer?

Chapter 12 Solutions

To find: minimum speed of a particle with trajectory

$c (t) = (t^{3}, t^{- 2}) for t \geq 0.5$ .