Explanation Given: Polar coordinates of a moving particle at time t are ( r ( t ) , θ ( t ) ) Formula used: (F1) Trigonemetric identity, ( cos 2 θ + sin 2 θ ) = 1 (F2) Rules of differentiation, A) d ( f g ) d x = f d ( g ) d x + g d ( f ) d x B) d ( f o g ) d x = d ( f ) d g d ( g ) d x Calculation: Polar coordinates of a moving particle at time t are ( r ( t ) , θ ( t ) ) , Cartesian coordinates and polar coordinates are related as follow, x ( t ) = r ( t ) cos θ ( t ) , y ( t ) = r ( t ) sin θ ( t ) For a given path, the time derivative gives the veleocity. Hence, taking the time derivative we get, x ˙ ( t ) = r ˙ ( t ) cos θ ( t ) − r ( t ) θ ˙ ( t ) sin θ ( t ) y ˙ ( t ) = r ˙ ( t ) sin θ ( t ) + r ( t ) θ ˙ ( t ) cos θ ( t ) …(using rules of differentiation) Now, speed ( s ) of the particle is the magnitude of the velocity vector (

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Chapter 12.4, Problem 43E

To determine

Polar coordinates of a moving particle at time $t$ are $(r (t), θ (t))$ ,

To prove that, the particle’s speed is,

$\sqrt{{(\frac{d r}{d t})}^{2} + r^{2} {(\frac{d θ}{d t})}^{2}}$

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Chapter 12.4, Problem 42E

Chapter 12.4, Problem 44E

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Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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Polar coordinates of a moving particle at time t are ( r ( t ) , θ ( t ) ) , To prove that, the particle’s speed is, ( d r d t ) 2 + r 2 ( d θ d t ) 2

Solution Summary: The author explains how to determine the polar coordinates of a moving particle at time t.

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Polar coordinates of a moving particle at time $t$ are $(r (t), θ (t))$ ,

To prove that, the particle’s speed is,

$\sqrt{{(\frac{d r}{d t})}^{2} + r^{2} {(\frac{d θ}{d t})}^{2}}$

Want to see the full answer?

Chapter 12 Solutions

Polar coordinates of a moving particle at time t are ( r ( t ) , θ ( t ) ) , To prove that, the particle’s speed is, ( d r d t ) 2 + r 2 ( d θ d t ) 2

Solution Summary: The author explains how to determine the polar coordinates of a moving particle at time t.

Concept explainers

Videos

Polar coordinates of a moving particle at time t are (r(t),θ(t)), To prove that, the particle’s speed is, (drdt)2+r2(dθdt)2

Want to see the full answer?

Chapter 12 Solutions

Polar coordinates of a moving particle at time $t$ are $(r (t), θ (t))$ ,

To prove that, the particle’s speed is,

$\sqrt{{(\frac{d r}{d t})}^{2} + r^{2} {(\frac{d θ}{d t})}^{2}}$