The acceleration of a particle moving along a straight line is inversely proportional to its speed with

the constant of porpotionality being k. The body starts from the position s = x0
^i with initial speed
v0.
Determine the velocity of the particle as a function of time t.
Determine the velocity of the particle as a function of position s

To determine the velocity of the particle as a function of time, we need to find an equation that relates acceleration, speed, and time. We know that acceleration is inversely proportional to speed with the constant of proportionality being k. Mathematically, this can be represented as:

a = k / v

Where:
a = acceleration
k = constant of proportionality
v = speed

Since we are given the initial speed v0, we can rewrite the equation as:

a = k / (v0 + kt)

Now, we can use the definition of acceleration to relate it to the rate of change of velocity with respect to time:

a = dv/dt

Rearranging the equation, we have:

dv/dt = k / (v0 + kt)

To solve this differential equation, we can separate the variables and integrate.

dv / (v0 + kt) = k dt

Integrating both sides gives:

ln|v0 + kt| = kt + C

Where C is the constant of integration. Now, we can exponentiate both sides:

|v0 + kt| = e^(kt + C)

Since the absolute value can be positive or negative, we can write:

v0 + kt = ±e^(kt + C)

Next, we need to consider the initial condition. At time t = 0, the particle has an initial velocity of v0. Using this condition, we can determine the constant of integration C:

v0 + k(0) = ±e^(k(0) + C)
v0 = ±e^C

Let's assume v0 is positive, so we can write:

v0 = e^C

Solving for C gives:

C = ln(v0)

Now we can rewrite the equation as:

v0 + kt = ±e^(kt + ln(v0))

Simplifying further, we have:

v(t) = -v0 * e^(-kt) or v(t) = v0 * e^(kt)

This is the velocity of the particle as a function of time.

To determine the velocity of the particle as a function of the position, we need to find a relationship between velocity, position, and time. We can start with the definition of velocity:

v = ds / dt

Where:
v = velocity
s = position
t = time

Rearranging the equation, we have:

ds = v dt

To relate position and time, we can integrate both sides:

∫ds = ∫v dt

This gives:

s - s0 = ∫v dt

Now, we can substitute the expression for v(t) we derived earlier:

s - s0 = ∫(±v0 * e^(kt)) dt

Integrating both sides gives:

s - s0 = ±(v0 / k) * e^(kt) + C

Simplifying further, we have:

s = s0 ± (v0 / k) * e^(kt) + C

Where C is the constant of integration.

This is the velocity of the particle as a function of position.