Formulas for constant power acceleration

[rcgldr]

This isn't homework. A friend asked about this so I decided to work out the formulas, but wanted to know if this was already done by someone here (otherwise I'll do the math).

p = power (constant)
a = acceleration
v = velocity
x = position
t = time
f = force

Assume an object is initially at rest, at position zero and time zero:

v₀ = 0
x₀ = 0
t₀ = 0

f = m a
p = f v
f = p / v

first step

a = f / m = dv/dt = p / (m v)
v dv = (p/m) dt
1/2 v² = (p/m) t

[tex] v = \frac{dx}{dt} = \sqrt {\frac{2\ p\ t}{m}} [/tex]

This is continued to find x as a function of t, then t as a function of x

Then determine f(x) = p / v(x)

and finally show that work done is

[tex]p\ t_1 = \int_0^{x_1} f(x) dx [/tex]

 

[mfb]

You can simply integrate your equation for v(t) and get x(t). That can be used to get v(x) and the other expressions.

 

[rcgldr]

You can simply integrate your equation for v(t) and get x(t). That can be used to get v(x) and the other expressions.

I know that, was just wondering if someone here had already done this in a previous thread. The previous threads I did find never actually completed the formulas. I'll go ahead and do this later.

 

[mfb]

I am sure this has been done before, it is a nice and easy problem in mechanics and can be solved with very basic concepts.

 

[rcgldr]

[tex] p = f \ v [/tex]
[tex] a = \frac{dv}{dt} = \frac{f}{m} = \frac {p} {m\ v} [/tex]
[tex] v\ dv = \frac{p}{m}\ dt [/tex]
[tex] \frac{1}{2}\ v^2 = \frac{p}{m}\ t [/tex]
[tex] v = \frac{dx}{dt} = \sqrt {\frac{2\ p\ t}{m}} [/tex]
[tex] dx = \sqrt {\frac{2\ p\ t}{m}}\ dt [/tex]
[tex] x = \sqrt {\frac{8\ p\ t^3}{9\ m}} [/tex]
[tex] t = \sqrt[3] {\frac{9\ m\ x^2}{8\ p}} [/tex]
a as a function of t:
[tex] a = \frac {p} {m\ v} = \frac {p} {m\ {\sqrt {\frac{2\ p\ t}{m}}}}
= \sqrt {\frac{p}{2\ m\ t}}[/tex]
v as function of x:
[tex] v = \sqrt {\frac{2\ p\ \sqrt[3] {\frac{9\ m\ x^2}{8\ p}}}{m}}
= \sqrt[3] {\frac{3\ p\ x}{m}} [/tex]
a as a function of x:
[tex] a = \frac{p}{m \ v} = \frac{p}{m \ \sqrt[3] {\frac{3\ p\ x}{m}}}
= \sqrt[3] {\frac{p^2}{3\ m^2\ x}} [/tex]
f as a function of x:
[tex] f = m\ a = \sqrt[3] {\frac{m\ p^2}{3\ x}} [/tex]
work done versus x:
[tex] w = \int_0^x \sqrt[3] {\frac{m\ p^2}{3\ x}} \ dx
= \sqrt[3]{\frac{9\ m\ p^2\ x^2}{8}} [/tex]
work done versus time:
[tex] w = \sqrt[3]{\frac{9\ m\ p^2\ \left( \sqrt {\frac{8\ p\ t^3}{9\ m}} \right )^2}{8}}
= p \ t [/tex]