# How Is Velocity Expressed as a Function of Position with Given Power Input?

• e2e8
In summary, the conversation discusses the power and velocity of a vehicle moving along a track. The power is represented by the formula P=d(1/2 m v^2)/dt and the velocity is represented by v(x). The formula for power is further manipulated to incorporate the variables of distance and velocity. It is suggested to use the formula e2dv/dt=dv/dx*dx/dt to solve the differential equation for v(x).
e2e8
Power to a vehicle moving along a track is alpha*x^1/2. No dissipation. alpha is constant x is distance from beginning of the track. zero initial velocity.
What is Velocity as function of position?

P=d KE/dt
P=d(1/2 m v^2)/dt
P=1/2 m d((dx/dt)^2)/dt
P=1/2 m 2 v dv/dt
alpha x^1/2 = 1/2 m 2 v dv/dt

And that is as far as I get. After some random manipulations I was not able to yield anything useful. Some assistance or guidance would be greatly appreciated. Thank you.

e2

dv/dt=dv/dx*dx/dt = dv/dx *v.

use this in your last formula then solve the differential equation for v(x).

ehild

=alpha*x^1/2
d(e2)/dt=d(1/2 m v^2)/dt
2*alpha*(dx/dt)*(d2x/dt2)=m*2v*(dv/dt)
alpha*dx/dt*d2x/dt2=m*v*dv/dt
alpha*dx/dt*d2x/dt2=alpha*x^1/2*dv/dt
dx/dt*d2x/dt2=x^1/2*dv/dt
d2x/dt2=x^-1/2*dv/dt
Integrating both sides:
dx/dt=x^-1/2*v + C1
Integrating again:
x=-2*v*x^1/2 + C1*t + C2
Since initial velocity is zero, C1=0.
Therefore, velocity as a function of position is:
v=-1/2*x^-1/2*x + C3
where C3 is the constant of integration.

## 1. What is the definition of "Power as function of position"?

Power as function of position is a measure of the rate at which work is done or energy is transferred based on an object's position. It takes into account the distance an object moves and the time it takes to move that distance.

## 2. How is "Power as function of position" calculated?

The formula for calculating power as function of position is P = W/t, where P is power in watts, W is work in joules, and t is time in seconds.

## 3. What is the difference between power and energy?

Power is the rate at which energy is transferred or work is done, while energy is the ability to do work. Power is measured in watts, while energy is measured in joules.

## 4. How does an object's position affect its power?

An object's position can affect its power by changing the distance it travels and the time it takes to travel that distance. The further an object moves in a given time, the higher its power will be.

## 5. What are some real-life applications of "Power as function of position"?

Power as function of position is used in various fields such as physics, engineering, and sports. It is used to calculate the power output of machines and vehicles, determine the efficiency of energy conversion, and analyze the performance of athletes in sports like running and cycling.

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