# Solve Maximum Velocity: Determine a, x, k, v for Particle

• Marioqwe
In summary, the problem involves a particle starting from rest at the origin with an acceleration of a=k/(x+4)^2, where a and x are in m/s2 and m, respectively, and k is a constant. The velocity of the particle is 4 m/s when x=8m. Part (a) and (b) were easily calculated, but there is difficulty with part (c) in finding the maximum velocity. To determine this, we set the acceleration equal to zero and integrate to find v as a function of x. From there, we can write v=dx/dt and find x as a function of time. Differentiating again and taking the limit as t goes to infinity will give us the maximum
Marioqwe

## Homework Statement

A particle starts from rest at the origin and is given an acceleration a=k/(x+4)^2, where a and x are expressed in m/s2 and m, respectively, and k is a constant. Knowing that the velocity of the particle is 4 m/s when x=8m, determine (a) the value of k, (b) the position of the particle when v = 4.5 m/s, (c) the maximum velocity of the particle.

## The Attempt at a Solution

I calculated part (a) and (b) easily. However, I am having some troubles with part (c).
The maximum velocity occurs when the acceleration is equal to zero because there is either a minimum or maximum right? So I set

0 = k/(x+4)^2

which cannot be.
What am I doing wrong?

write:

$$v\frac{dv}{dx}=\frac{k}{(x+4)^{2}}$$

Integrate to get v as a function of x and then write v=dx/dt so find x as a function of t then differentiate again and examine the limit as t tends to infinity.

I am getting a very ugly integral. But you said to get v as function of x right?
Then

v = dx/dt ---> dt = (1/v)dx

and integrate the above to get the position x as a function of time? And then take the limit of that function as t goes to infinity? Why would that give me the maximum velocity? Is it because the graph of x(t) has a vertical asymptote?

What do you get for the integral?

## 1. How do you calculate maximum velocity?

Maximum velocity can be calculated by finding the derivative of the position function and setting it equal to zero. This will give the value of the independent variable at which the slope of the function is at its maximum.

## 2. What do a, x, k, and v represent in the equation for maximum velocity?

In the equation for maximum velocity, a represents the acceleration, x represents the position, k represents the spring constant, and v represents the velocity. These variables are important in determining the maximum velocity of a particle.

## 3. How is the acceleration of a particle related to its maximum velocity?

The acceleration of a particle is directly related to its maximum velocity. As the acceleration increases, the maximum velocity will also increase. Conversely, if the acceleration decreases, the maximum velocity will also decrease.

## 4. Can the maximum velocity of a particle change over time?

Yes, the maximum velocity of a particle can change over time. This can occur if there are changes in the acceleration or other factors that affect the particle's motion.

## 5. Why is it important to determine the maximum velocity of a particle?

Determining the maximum velocity of a particle is important because it can give insight into the particle's motion and how it will behave in a given situation. It can also help in predicting potential dangers or impacts in systems that involve moving particles.

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