Question

Theory and ExamplesThe equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.s = 2 − 2 sin t

Accepted Answer

To find the body's velocity at time $$ t = \frac{\pi}{4} $$, we need to take the first derivative of the position function $$ s(t) = 2 - 2 \sin t $$ with respect to $$ t . $$The derivative of $$ -2 \sin t  is$$ $$ -2 \cos t . $$Therefore, the velocity function $$ v(t)  is$$ $$ v(t) = -2 \cos t . $$Next, to find the speed at $$ t = \frac{\pi}{4} $$, we take the absolute value of the velocity function at that time. Speed is the magnitude of velocity, so $$ 	ext{speed} = |v(t)| = |-2 \cos(\frac{\pi}{4})| . To $$find the acceleration at $$ t = \frac{\pi}{4} $$, we take the derivative of the velocity function $$ v(t) = -2 \cos t . $$The derivative of $$ -2 \cos t  is$$ $$ 2 \sin t . $$Thus, the acceleration function $$ a(t)  is$$ $$ a(t) = 2 \sin t . $$For the jerk, which is the derivative of acceleration, we differentiate the acceleration function $$ a(t) = 2 \sin t . $$The derivative of $$ 2 \sin t  is$$ $$ 2 \cos t . $$Therefore, the jerk function $$ j(t)  is$$ $$ j(t) = 2 \cos t . $$Finally, evaluate each of these functions at $$ t = \frac{\pi}{4} $$: $$ v(\frac{\pi}{4}) = -2 \cos(\frac{\pi}{4}) $$, $$ 	ext{speed} = |-2 \cos(\frac{\pi}{4})| $$, $$ a(\frac{\pi}{4}) = 2 \sin(\frac{\pi}{4}) $$, and $$ j(\frac{\pi}{4}) = 2 \cos(\frac{\pi}{4}) .$$

Theory and Examples

The equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.

s = 2 − 2 sin t

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주요 개념

Differentiation

Velocity and Speed

Higher Order Derivatives