Яндекс.Метрика

Monday, October 17, 2016

Solve m₁g-t=m₁a₁, m₂g-½t=m₂a₂, m₃g-½t=m₃a₃, -2a₁=a₂+a₃, for t,a₁,a₂,a₃ where t is a tension of the cord, which is passing through the pulley A


Solve 
m₁g-t=m₁a₁, 
m₂g-½t=m₂a₂, 
m₃g-½t=m₃a₃, 
-2a₁=a₂+a₃, 
for t,a₁,a₂,a₃
where t is a tension of the cord, which is passing through the pulley A

Saturday, October 1, 2016

Constant Power

What is the power output of an engine when a car of mass M accelerates with the constant power from rest to a final speed V over a distance d on flat, level ground?

Ignore energy lost due to friction and air resistance.
Derive a formula for the power P in terms of variables: the mass M, the final speed V, and the distance d.
P
=P(M,V,d) ?
v=v(t) The speed is a function of time.
Pt = ½m v(t)² Because of the energy conservation law.
v(t)²=2tP/m=(2P/m)t
v(t)=(2P/m)½t½
x(t)=(2P/m)½ (⅔t3/2)
x(t)²=(2P/m) (⅔)² t³
t=v(t)² / (2P/m)
t³=v(t)⁶ / (2P/m
x(t)²=(2P/m) (⅔)² t³=(2P/m) (⅔)² v(t)⁶ / (2P/m
x(t)²=(⅔)² v(t)⁶ / (2P/m) ²
x(t)=(⅔) v(t)³ / (2P/m)
2P/m = (⅔) v(t)³ / x(t)
P = ½(⅔) v(tm / x(t)

P = ⅓ v(tm / x(t) = ⅓ V³m / d

Wednesday, September 21, 2016

What is the ratio of the speed of the rocket when it has covered half the distance to the star to its speed when it has traveled half the time necessary to reach Alpha Centauri?

A rocket, speeding along toward Alpha Centauri, has an acceleration a(t) = At². Assume that the rocket began at rest at the Earth (x = 0) at t = 0. Assuming it simply travels in a straight line from Earth to Alpha Centauri (and beyond), what is the ratio of the speed of the rocket when it has covered half the distance to the star to its speed when it has traveled half the time necessary to reach Alpha Centauri?

a(t) = At²
v(0) = 0
x(0) = 0
D = x(T)
D / 2 = x(τ)
v(τ) / v(½T) = ?

Solution
a(t) = At²
v(t) = At³
x(t) = At⁴ / 12
D = AT⁴ / 12
T⁴ = 12D / A
t⁴ = 12x / A
τ⁴ = 12(D / 2) / A
τ = 12D / A · (½) = (½)T⁴
τ = √(½) · T
x(τ) = Aτ⁴ / 12 = A(√(½) · T)⁴ / 12 = ½ AT⁴ / 12 = ½D

v(½T) = ⅓A(½T)³ = (½)³ · ⅓AT³
v(τ) = v(⁴√(½) · T) = ⅓A(⁴√(½) · T)³ = (⁴√(½))³ · ⅓AT³

v(τ) / v(½T) = {(⁴√(½))³ · ⅓AT³} / {(½)³ · ⅓AT³} = {(⁴√(½))³ } / {(½)³} = (2 / ⁴2)³ = 8 / ⁴8 = (8)³

Tuesday, July 26, 2016

Vector Sum


Задача на развитие интуитивного, образного понимания векторов и их свойств и, одновременно, на развитие глазомера. Сложите шесть изоброженных радиус-векторов. Найдите номер пятна, который достигает голова вектора-суммы. Начало вектора суммы, как и исходных векторов, расположено на пятне с номером 0?

Wednesday, July 13, 2016

Find functions with the following properties

Find functions with the following properties
Найдите функции, обладающие следующими свойствами.
For example f₁₁(x) = kx where k is a constant.
Например, f₁₁(x) = kx, где k - любая константа.
‪#‎function‬ ‪#‎properties‬ ‪#‎функции‬ ‪#‎свойства‬ ‪#‎constant‬ ‪#‎константа‬

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