Calculate the acceleration vector given the velocity function in unit vector notation.
Describe the motion of a particle with a constant acceleration in three dimensions.
Use the one-dimensional motion equations along perpendicular axes to solve a problem in two or three dimensions with a constant acceleration.
Express the acceleration in unit vector notation.
Instantaneous acceleration
In addition to obtaining the displacement and velocity vectors of an object in motion, we often want to know its
acceleration vector at any point in time along its trajectory. This acceleration vector is the instantaneous acceleration and it can be obtained from the derivative with respect to time of the velocity function, as we have seen in a previous chapter. The only difference in two or three dimensions is that these are now vector quantities. Taking the derivative with respect to time
we find
The acceleration in terms of components is
Also, since the velocity is the derivative of the position function, we can write the acceleration in terms of the second derivative of the position function:
Finding an acceleration vector
A particle has a velocity of
(a) What is the acceleration function? (b) What is the acceleration vector at
t = 2.0 s? Find its magnitude and direction.
Solution
(a) We take the first derivative with respect to time of the velocity function to find the acceleration. The derivative is taken component by component:
(b) Evaluating
gives us the direction in unit vector notation. The magnitude of the acceleration is
Significance
In this example we find that acceleration has a time dependence and is changing throughout the motion. Let’s consider a different velocity function for the particle.
A particle has a position function
(a) What is the velocity? (b) What is the acceleration? (c) Describe the motion from
t = 0 s.
Strategy
We can gain some insight into the problem by looking at the position function. It is linear in
y and
z , so we know the acceleration in these directions is zero when we take the second derivative. Also, note that the position in the
x direction is zero for
t = 0 s and
t = 10 s.
Solution
(a) Taking the derivative with respect to time of the position function, we find
The velocity function is linear in time in the
x direction and is constant in the
y and
z directions.
(b) Taking the derivative of the velocity function, we find
The acceleration vector is a constant in the negative
x -direction.
(c) The trajectory of the particle can be seen in
[link] . Let’s look in the
y and
z directions first. The particle’s position increases steadily as a function of time with a constant velocity in these directions. In the
x direction, however, the particle follows a path in positive
x until
t = 5 s, when it reverses direction. We know this from looking at the velocity function, which becomes zero at this time and negative thereafter. We also know this because the acceleration is negative and constant—meaning, the particle is decelerating, or accelerating in the negative direction. The particle’s position reaches 25 m, where it then reverses direction and begins to accelerate in the negative
x direction. The position reaches zero at
t = 10 s.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .