Answer to Question #188039 in Calculus for Daniel

Question

You have been given an equation to calculate velocity of a particle after t (time,

s) and passing a fixed point (P). the equation is:

v(t) = 3 Cos(πt) − 2Sin(πt) (Eq.1)

● v(t) is the instantaneous velocity of the car (m/s)

● t is the time in seconds

Your tasks is:

a) Define the given engineering problem (Eq. 1) and find the

application to use it.

b) Identify what type of calculus methods and mathematical model

can be sued.

c) Analyse the given engineering problem (Eq. 1) and break the

problem down into a series of manageable elements.

d) Explain what type of data and variables can be calculated

(extracted) from a given equation using calculus method and

mathematical method.

e) Explain the reasons for each element (why we used these

methods).

f) How the given engineering problem (Eq.1) can be solved

accurately using calculus methods and mathematical model.

Answer 1

$v(t)=$ 3cos$({\pi}t)-2sin({\pi}t)$ ...........(1}

$\to$ v(t) is the instantaneous velocity of the car (m/s)

$\to$ t is the time in seconds

A) The given engineering problem is about instantaneous velocity of car at any time t since by the equation we can say that both$\ sin({\pi}t)$ and $\cos({\pi}t)$ terms are present we can say that motion of car is not linear it is a harmonic motion of the car.

The application to use it is very simple by putting t=0,1,2,3...so on in the equation we get instantaneous velocity of particle at time t=0,1,2,3....$we \ know \ that\ cos(n{\pi})=0$ where n=0,1,2,3,....etc

B)The calculus method that can be used here is by putting V=$\dfrac{dx}{dt}$ ,and than by integerating both side of the equation we can get instantaneous equation of a particle covered by the particle during the whole journey and the mathematical model that can be used is that we can put t=0 and can get instantaneous distance of particle at any time.

C) By analysing the given engineering problem we can identify that clearly that we can get instantaneous velocity, instantaneous acceleration, instantaneous speed....we can also get where the car has maximum speed and the car has minimum speed .

D)By the help of the equation (1) we can calculate instantaneous acceleration of car ,instantaneous speed, we can also calculate where the velocity of particle is maximum, where the velocity of particle is minimum...we can also get the time period of car by using simple integeration and differentiation we can get the above things easily.

E) By putting V=$\dfrac{dx}{dt}$ and integrating on both side we can get instantaneous distance of car and by $\dfrac{dv}{dt}$ we can get instantaneous acceleration of car .

Instantaneous distance Equation of car can be given as $x(t)=\dfrac{3sin({\pi}t)}{\pi}+\dfrac{2cos({\pi}t)}{\pi}$

instantaneous aceleration of car can be given as a(t)=$-3{\pi}sin({\pi}t)-2{\pi}cos({\pi}t).$

F)The given engineering problem can be solved very simply

$v(t)=$ 3cos$({\pi}t)-2sin({\pi}t)$,

Now putting any value of t=0,1,2,3...by putting any value we can get car velocity at any position .

x(t)=$\dfrac{3sin{(\pi}t)}{\pi}+\dfrac{2cos{(\pi}t)}{\pi}$

​Now putting t=0,1,2,3,...by putting these value we can get instantaneous distance covered by car easily.

a(t)=$-3{\pi}sin({\pi}t)-2{\pi}cos({\pi}t).$

Now putting t=0,1,2,3...by putting these value we can get instantaneous aceleartion of car easily.