9 ## INSTANTANEOUS VELOCITY## AND## RELATED RATESThe definition of instantaneous velocity ONE OF THE most important applications of calculus is to motion in a straight line, which is called rectilinear motion. Consider a particle moving in a straight line. Let
When we know Now, if the particle moves with constant velocity -- which is called uniform motion -- then we don't need calculus for that. In other words, if the equation of motion is
then at every instant of time, the velocity is 22 m/sec. For, the slope of that line, which is 22, is rate of change of In each 1 second of time, the particle moves a distance of 22 meters.
That is not a realistic picture, of course, because at 0 seconds the velocity is surely not 22 meters/sec There must have been an to that constant velocity. During that acceleration, the velocity was not constant. The graph was not a straight line. The definition of instantaneous velocity For any equation of motion
The instantaneous velocity is the value of the slope of the tangent line at Example 1. Let the following be the equation of motion:
Let a) What is the instantaneous velocity at 10:05 AM?
b) What is the instantaneous velocity at 10:15 AM?
Instantaneous velocity is very different from ordinary velocity, which, to calculate, requires an interval of time. "Instantaneous velocity," like any limit, is defined at an instant of time. It is purely logical, and can never be observed or measured. To measure a velocity, it is necessary to know both a distance Δ A body in motion is in motion during every interval of time in which it moves. If we say at every instant of time -- and assume that time is composed of instants -- then we invite the arrow paradox of Zeno. At every instant, he argued, there is no change of time, no change of position, which means: no motion. The definition of instantaneous velocity does not imply that time is composed of instants. It defines how to evaluate that velocity at any Problem 1. It has been found by experiment that a body falling from rest under the influence of gravity, follows approximately this equation of motion:
a) At the end of 3 seconds, how far has the body fallen? To see the answer, pass your mouse over the colored area.
b) What is its instantaneous velocity at the end of 3 seconds?
The second derivative The derivative of
Consider this equation of motion,
Then the first derivative is the velocity
The second derivative is the rate of change of the velocity with respect to time. That is called the acceleration
If Problem 2. A body moves in a straight line according to this equation of motion:
where a) What is its position at the end of 5 sec?
b) What is the equation for its velocity
c) What is its velocity
d) What is the equation for its acceleration
e) What is its acceleration at the end of 5 seconds?
The acceleration is constant. Problem 3. Under the influence of gravity, a body moves according to this equation of motion:
a) What is the physical significance of the constant
It is the body's initial position, b) How fast is the body moving after 5 seconds?
c) What is the physical significance of the constant
approximately 9.8 m/sec². Related rates Example 2. a) If the radius of a circle is expanding, write the equation that shows
b) If the radius is expanding at the rate of 2 cm/min, how fast is the area
Example 3. A boy is walking at the rate of 5 miles per hour toward the foot of a flag pole 60 feet high. At what rate is his distance from the top of the pole changing when he is 80 feet from its foot?
Let the boy be at the point
The figure is a right triangle. Therefore,
Differentiate implicitly with respect to
Therefore,
minus sign because Therefore,
Problem 4. The side of a square is
2
Problem 5. The side of an equilateral triangle is
½ Problem 6. a) The surface area If
If
Problem 7. The base and height of a rectangle are
A = According to the product rule:
Problem 8. a) A ladder 50 feet long is leaning against a wall. If the foot of the ladder
b) If the foot is being pulled away at the rate of 3 ft/min: 1) how fast is the top descending when the foot is 14 feet from the 7/8 ft/min. 2) when is the top descending at the rate of 4 ft/min? When the bottom is 40 feet from the wall. c) When will the top and bottom move at the same rate? When the bottom is 25 feet from the wall.
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