# Find the equation for velocity as a function of time

• ggb123
In summary, the equation for velocity as a function of time for a rock sliding on a horizontal surface with a retarding force proportional to the square root of its velocity is v = v_o - (\frac{k}{m}t)/2 + (\frac{k^2t^2}{4m^2}).
ggb123

## Homework Statement

A rock with mass "m" slides with initial velocity on a horizontal surface. A retarding force F that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock: F = -k * v^(0.5)

Find the equation for velocity as a function of time.

F = ma
a = dv/dt

## The Attempt at a Solution

I'm using online software and used up all my attempts at the answer, so it gave it to me:
vi = initial velocity
v = vi - ((vi)^0.5 * k * t)/m + (k^2 * t^2)/(4m^2)

And I have no idea how to get this. I'm thinking this is because I don't know much about integration. I'm hoping somebody could help me out in explaining this to me, even just a brief outline of the steps to the final answer would be great.

Any help is really appreciated, thanks

$$F = -kv^{1/2}$$

$$a = \frac{dv}{dt} = -\frac{k}{m}v^{1/2}$$

$$\frac{dv}{v^{1/2}} = -\frac{k}{m}dt$$

Taking integration you get

$$2v^{1/2} = -\frac{k}{m}t + C$$

When t = 0, $$C = 2v_o^{1/2}$$

$$2v^{1/2} = -\frac{k}{m}t + 2v_o^{1/2}$$

Square both the sides and simplify.

## 1. What is the formula for velocity as a function of time?

The formula for velocity as a function of time is v(t) = v0 + at, where v0 is the initial velocity, a is the acceleration, and t is the time.

## 2. How do you find the equation for velocity as a function of time?

The equation for velocity as a function of time can be found by using the basic formula v = d/t and solving for v. This gives the equation v = Δd/Δt, which can be rewritten as v = (d2 - d1)/(t2 - t1). This equation can then be simplified to v = d/t if the acceleration is constant.

## 3. What is the difference between velocity and speed?

Velocity and speed are often used interchangeably, but they have different meanings in physics. Velocity is a vector quantity that describes the rate of change of an object's position with respect to time, including its direction. Speed, on the other hand, is a scalar quantity that only describes how fast an object is moving, regardless of direction.

## 4. How does acceleration affect the equation for velocity as a function of time?

Acceleration affects the equation for velocity as a function of time by adding a term for acceleration (a) multiplied by the time (t). This is represented in the formula v(t) = v0 + at. When acceleration is constant, the term for acceleration can be simplified to a single value, making the equation v(t) = v0 + at.

## 5. Can the equation for velocity as a function of time be used for non-uniform acceleration?

Yes, the equation for velocity as a function of time can be used for non-uniform acceleration. In this case, the acceleration (a) would be changing over time, so the equation would need to be integrated to find the velocity as a function of time. This integration can be done using calculus.

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