We discussed air resistance at the beginning of this section. The next example shows how to apply this concept for a ball in vertical motion. Other factors can affect the force of air resistance, such as the size and shape of the object, but we ignore them here.
A ball with air resistance
A racquetball is hit straight upward with an initial velocity of
m/s. The mass of a racquetball is approximately
kg. Air resistance acts on the ball with a force numerically equal to
where
represents the velocity of the ball at time
Find the velocity of the ball as a function of time.
How long does it take for the ball to reach its maximum height?
If the ball is hit from an initial height of
meter, how high will it reach?
The mass
and
The initial velocity is
m/s. Therefore the initial-value problem is
Dividing the differential equation by
gives
The differential equation is linear. Using the problem-solving strategy for linear differential equations:
Step 1. Rewrite the differential equation as
This gives
and
Step 2. The integrating factor is
Step 3. Multiply the differential equation by
Step 4. Integrate both sides:
Step 5. Solve for
using the initial condition
Therefore the solution to the initial-value problem is
The ball reaches its maximum height when the velocity is equal to zero. The reason is that when the velocity is positive, it is rising, and when it is negative, it is falling. Therefore when it is zero, it is neither rising nor falling, and is at its maximum height:
Therefore it takes approximately
second to reach maximum height.
To find the height of the ball as a function of time, use the fact that the derivative of position is velocity, i.e., if
represents the height at time
then
Because we know
and the initial height, we can form an initial-value problem:
Integrating both sides of the differential equation with respect to
gives
The weight of a penny is
grams (United States Mint, “Coin Specifications,” accessed April 9, 2015, http://www.usmint.gov/about_the_mint/?action=coin_specifications), and the upper observation deck of the Empire State Building is
meters above the street. Since the penny is a small and relatively smooth object, air resistance acting on the penny is actually quite small. We assume the air resistance is numerically equal to
Furthermore, the penny is dropped with no initial velocity imparted to it.
Set up an initial-value problem that represents the falling penny.
Solve the problem for
What is the terminal velocity of the penny (i.e., calculate the limit of the velocity as
approaches infinity)?
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
