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**Velocity, Acceleration, Jerk**

Section 3.4a

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**Definition: Instantaneous Rate of Change**

The difference quotient: When we let h approach 0, we saw the rate at which a function was changing at a particular point x… Definition: Instantaneous Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative provided the limit exists.

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**Example 1: Enlarging Circles**

(a) Find the rate of change of the area A of a circle with respect to its radius r. (b) Evaluate the rate of change of A at r = 5 and at r = 10. (c) If r is measured in inches and A is measured in square inches, what units would be appropriate for dA/dr ? Instantaneous rate of change of A with respect to r : Units? Rate at r = 5: Square inches (of area) per inch (of radius) Rate at r = 10: The rate of change gets bigger as r gets bigger!!!

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Motion Along a Line If an object is moving along an axis, we may know its position s on that line as a function of time t : The displacement of the object over the interval t to t + t : The average velocity of the object over this time interval: How would we find the object’s velocity at the exact instant t ?

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**Definition: Instantaneous Velocity**

The (instantaneous) velocity is the derivative of the position function s = f(t) with respect to time. At time t the velocity is Definition: Speed Speed is the absolute value of velocity. Speed =

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**Example 2: Reading a Velocity Graph**

A particle moves along an axis, and its velocity is shown in the graph below. When does the particle have maximum speed ? v (m/sec) The particle moves forward for the first 10 seconds, then moves backward for the next 8 seconds, stands still for 4 seconds, and then moves forward again. The particle achieves its maximum speed at about t = 15, while moving backward. 5 t (sec)

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**Definition: Acceleration**

Acceleration is the derivative of velocity with respect to time. If a body’s velocity at time t is v(t) = ds/dt, then the body’s acceleration at time t is Definition: Jerk Jerk is the derivative of acceleration with respect to time. If a body’s position at time t is s(t), the body’s jerk at time t is

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**Free-Fall Constants (Earth)**

English units: (s in feet) Metric units: (s in meters)

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**Example 3: Modeling Vertical Motion**

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of after t seconds. (a) How high does the rock go? Model the situation: s In our model, velocity is positive on the way up, and negative on the way down. s v = 0 max Find velocity at any time t: Height (ft) ft/sec The velocity is zero when: sec s = 0

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**Example 3: Modeling Vertical Motion**

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of after t seconds. (a) How high does the rock go? Model the situation: s The maximum height of the rock is the height at t = 5 sec: s v = 0 max Height (ft) ft s = 0

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**Example 3: Modeling Vertical Motion**

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of after t seconds. (b) What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? s s v = 0 max 256 t = ? Height (ft) ft/sec ft/sec s = 0 At both instants, the speed of the rock is 96 ft/sec

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**Example 3: Modeling Vertical Motion**

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of after t seconds. s (c) What is the acceleration of the rock at any time t during its flight? s v = 0 max 256 t = ? Height (ft) The acceleration is always negative!!! s = 0

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**Example 3: Modeling Vertical Motion**

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of after t seconds. s (d) When does the rock hit the ground? s v = 0 max 256 t = ? Height (ft) The rock hits the ground 10 seconds after the blast. Let’s graph the position, velocity, and acceleration functions together in the same window: [0,10] by [–160,400]. s = 0

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**Example 4: A Moving Particle**

A particle moves along a line so that its position at any time is given by the function where s is measured in meters and t is measured in seconds. (a) Find the displacement during the first 5 seconds. (b) Find the average velocity during the first 5 seconds. (c) Find the instantaneous velocity when t = 4. (d) Find the acceleration of the particle when t = 4. (e) At what values of t does the particle change direction? (f) Where is the particle when s is a minimum?

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**Example 4: A Moving Particle**

A particle moves along a line so that its position at any time is given by the function where s is measured in meters and t is measured in seconds. (a) Find the displacement during the first 5 seconds. Displacement = (b) Find the average velocity during the first 5 seconds. Average Velocity = (c) Find the instantaneous velocity when t = 4.

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**Example 4: A Moving Particle**

A particle moves along a line so that its position at any time is given by the function where s is measured in meters and t is measured in seconds. (d) Find the acceleration of the particle when t = 4. Acceleration = (e) At what values of t does the particle change direction? (f) Where is the particle when s is a minimum? Since acceleration is always positive, the position s is at a minimum when the particle changes direction, at t = 3/2.

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