4. Applications of Derivatives

Motion Analysis

4. Applications of Derivatives

Motion Analysis

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Multiple Choice

Given below is the graph of velocity with respect to time. At which time(s) would acceleration be 0?

At $t=0$

At $t=1$ & $t=3$

At $t=0$ & $t=5$

At $t=1$ , $t=2$ & $t=3$

Verified step by step guidance

To determine when acceleration is zero, we need to find when the velocity graph has a horizontal tangent, meaning the slope of the velocity graph is zero.

Examine the graph and identify points where the velocity curve is flat, indicating a zero slope. These points are where the graph changes direction or has a peak or trough.

On the graph, observe that at t=1, t=2, and t=3, the velocity curve has horizontal tangents. These are the points where the slope of the velocity graph is zero, indicating zero acceleration.

At t=1, the graph reaches a local minimum, at t=2, it has a local maximum, and at t=3, it reaches another local maximum. These points correspond to zero acceleration.

Thus, the times at which acceleration is zero are t=1, t=2, and t=3, as these are the points where the velocity graph has horizontal tangents.

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Textbook Question

Terminal velocity Refer to Exercises 95 and 96.

d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

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Multiple Choice

Given the position equation $s\left(t\right)$ , calculate the average velocity (in meters per second) based on the given interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\left(t\right)=9t-t^2$ , $0\le t\le3$

Given the position equation $s\left(t\right)$ , calculate the average velocity (in meters per second) based on the given time interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\left(t\right)=\frac{30}{t+5}$ , $-4\le t\le0$

Given the position of an object $s\left(t\right)=12t-t^2$ (in meters), find the acceleration of the object at $t=5$ seconds.

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Textbook Question

Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

a. Graph the position function.

$f (t) = 6 t^{3} + 36 t^{2} - 54 t; 0 \leq t \leq 4$

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Textbook Question

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y = 10 e^{- \frac{t}{2}} \cos (\frac{π t}{8})$ .

a. Graph the displacement function.

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Textbook Question

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the velocity and acceleration of the object at t = 1.

f(t) = t² − 4t; 0 ≤ t ≤ 5

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Textbook Question

Determine the acceleration of the object when its velocity is zero.

f(t) = t² - 4t; 0 ≤ t ≤ 5

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