1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
jolli1 [7]
3 years ago
6

A body of constant mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k|v|, whe

re k is a constant. Neglect changes in the gravitational force. a. Find the maximum height xm attained by the body and the time tm at which this maximum height is reached.
Mathematics
1 answer:
bixtya [17]3 years ago
3 0

Answer:

tm = tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] }

Xm = Xₐ = (v₀m)/k - ({m²g}/k²) ㏑(1+{kv₀/mg})

Step-by-step explanation:

Note, I substituted maximum time tm = tₐ and maximum height Xm = Xₐ

We will use linear ordinary differential equation (ODE) to solve this question.

Remember that Force F = ma in 2nd Newton law, where m is mass and a is acceleration

Acceleration a is also the rate of change in velocity per time. i.e a=dv/dt

Therefore F = m(dv/dt) = m (v₂-v₁)/t

There are two forces involved in this situation which are F₁ and F₂, where F₁ is the gravitational force and F₂ is the air resistance force.

Then, F = F₁ + F₂ = m (v₂-v₁)/t

F₁ + F₂ = -mg-kv = m (v₂-v₁)/t

Divide through by m to get

-g-(kv/m) = (v₂-v₁)/t

Let (v₂-v₁)/t be v¹

Therefore, -g-(kv/m) = v¹

-g = v¹ + (k/m)v --------------------------------------------------(i)

Equation (i) is a inhomogenous linear ordinary differential equation (ODE)

Therefore let A(t) = k/m and B(t) = -g --------------------------------(ia)

b = ∫Adt

Since A = k/m, then

b = ∫(k/m)dt

The integral will give us b = kt/m------------------------------------(ii)

The integrating factor will be eᵇ = e ⁽<em>k/m</em>⁾

The general solution of velocity at any given time is

v(t) = e⁻⁽b⁾ [ c + ∫Beᵇdt ] --------------------------------------(iiI)

substitute the values of b, eᵇ, and B into equation (iii)

v(t) = e⁻⁽kt/m⁾ [ c + ∫₋g e⁽kt/m⁾dt ]

Integrating and cancelling the bracket, we get

v(t) = ce⁻⁽kt/m⁾ + (e⁻⁽kt/m⁾ ∫₋g e⁽kt/m⁾dt ])

v(t) = ce⁻⁽kt/m⁾ - e⁻⁽kt/m⁾ ∫g e⁽kt/m⁾dt ]

v(t) = ce⁻⁽kt/m⁾ -mg/k -------------------------------------------------------(iv)

Note that at initial velocity v₀, time t is 0, therefore v₀ = v(t)

v₀ = V(t) = V(0)

substitute t = 0 in equation (iv)

v₀ = ce⁻⁽k0/m⁾ -mg/k

v₀ = c(1) -mg/k = c - mg/k

Therefore c = v₀ + mg/k  ------------------------------------------------(v)

Substitute equation (v) into (iv)

v(t) = [v₀ + mg/k] e⁻⁽kt/m⁾ - mg/k ----------------------------------------(vi)

Now at maximum height Xₐ, the time will be tₐ

Now change V(t) as V(tₐ) and equate it to 0 to get the maximum time tₐ.

v(t) = v(tₐ) = [v₀ + mg/k] e⁻⁽ktₐ/m⁾ - mg/k = 0

to find tₐ from the equation,

[v₀ + mg/k] e⁻⁽ktₐ/m⁾ = mg/k

e⁻⁽ktₐ/m⁾ = {mg/k] / [v₀ + mg/k]

-ktₐ/m = ㏑{ [mg/k] / [v₀ + mg/k] }

-ktₐ = m ㏑{ [mg/k] / [v₀ + mg/k] }

tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] }

Therefore tₐ = -m/k ㏑{ [mg/k] / [v₀ + mg/k] } ----------------------------------(A)

we can also write equ (A) as tₐ = m/k ㏑{ [mg/k] [v₀ + mg/k] } due to the negative sign coming together with the In sign.

Now to find the maximum height Xₐ, the equation must be written in terms of v and x.

This means dv/dt = v(dv/dx) ---------------------------------------(vii)

Remember equation (i) above  -g = v¹ + (k/m)v

Given that dv/dt = v¹

and -g-(kv/m) = v¹

Therefore subt v¹ into equ (vii) above to get

-g-(kv/m) = v(dv/dx)

Divide through by v to get

[-g-(kv/m)] / v = dv / dx -----------------------------------------------(viii)

Expand the LEFT hand size more to get

[-g-(kv/m)] / v = - (k/m) / [1 - { mg/k) / (mg/k + v) } ] ---------------------(ix)

Now substitute equ (ix) in equ (viii)

- (k/m) / [1 - { mg/k) / (mg/k + v) } ] = dv / dx

Cross-multify the equation to get

- (k/m) dx = [1 - { mg/k) / (mg/k + v) } ] dv --------------------------------(x)

Remember that at maximum height, t = 0, then x = 0

t = tₐ and X = Xₐ

Then integrate the left and right side of equation (x) from v₀ to 0 and 0 to Xₐ respectively to get:

-v₀ + (mg/k) ㏑v₀ = - {k/m} Xₐ

Divide through by - {k/m} to get

Xₐ = -v₀ + (mg/k) ㏑v₀ / (- {k/m})

Xₐ = {m/k}v₀ - {m²g}/k² ㏑(1+{kv₀/mg})

Therefore Xₐ = (v₀m)/k - ({m²g}/k²) ㏑(1+{kv₀/mg}) ---------------------------(B)

You might be interested in
Twice a number is 12 more than 5 times the number.
stira [4]

Answer:

  • -4

Step-by-step explanation:

<u>Let the number be x:</u>

  • 2x = 5x + 12
  • 2x - 5x = 12
  • -3x = 12
  • x = -4

It is -4

5 0
3 years ago
Solve for r write the answer in simplified form<br><br> 7 = - 7r
alex41 [277]

Answer:

r = -1

Step-by-step explanation:

To get r isolated, you have to divide each side by -7

7 divided by -7 is -1, therefore r = -1

3 0
3 years ago
PLEASE ITS NOT THAT HARD
nevsk [136]

Answer:

25.   0.0036    26.    1/64

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
The social studies teacher wants to know whether the students in the entire school prefer a model United Nations activity or a s
Vladimir79 [104]

All students in each grade because it should not be limited based on gender or if they like social studies or not.

7 0
3 years ago
Work out the size of angle ABC and give a reason for your answer
Mariulka [41]

Answer:

there is no picture so it is impossible to answer

Step-by-step explanation:

3 0
3 years ago
Other questions:
  • Three and one third divided by six
    5·1 answer
  • A bag has four green marbles, three red marbles, and three yellow marbles. What is the probability that you pick a red marble, d
    12·2 answers
  • The formula I = p r t gives the amount of simple interest (I) earned by investing principle ( p ) for rate ( r ) over a period o
    5·1 answer
  • A company that makes​ hair-care products had 7,000 people try a new shampoo. Of the 7,000 people, 28 had a mild allergic reactio
    12·1 answer
  • At Walt Disney World, 2 pretzels and 3 cans of soda cost $15.50. Four pretzels and five cans of soda cost $28.50. Which system o
    9·1 answer
  • What are m and b in the linear equation 12 + 6x + 15 + 10x = y?
    9·1 answer
  • Sand will be placed under the base of a circular pool with a diameter of 20 feet. 1 bag of sand covers about 6 square feet. How
    10·1 answer
  • The chart below describes the commissions earned by four different sales people. Sales Person Commission Earned Time on Sales Fl
    13·1 answer
  • In a right triangle, one of the angles measures 29º. What is the measure of the
    14·1 answer
  • Determine the intercepts of the line.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!