Sliding object against air resistance, working out when object comes to rest

[dbatten]

Homework Statement

A sledge of mass m = 85 kg has initial velocity v0 = 45 m/s and is moving horizontally along a straight line. The net force acting on the sledge is solely due to air resistance and is given by F = −k v2, where the constant is given by k = 1.2 kg/m. Find where the sledge will come to a stand still.

Homework Equations

I worked out that the velocity is given as v=(mv0)/(m+tkv0).

The Attempt at a Solution

I tried setting the velocity=0 but obviously that gives no answers. I then tried to differentiate the velocity to get the acceleration but I don't think this is the right route to go down.

 

[mark726]

Thank you for your post. I understand your confusion and I'm happy to help you find the solution to this problem.

Firstly, I want to clarify that the equation you have derived for the velocity is correct. However, in order to find the point where the sledge will come to a standstill, we need to set the velocity to 0 and solve for the time (t) it takes for this to happen.

Therefore, we have:

0 = (m*v0)/(m + t*k*v0)

Multiplying both sides by (m + t*k*v0), we get:

0 = m*v0

This means that the sledge will come to a standstill when the time (t) is equal to infinity. In other words, it will never come to a complete stop due to the constant force of air resistance acting on it.

I hope this clarifies your doubts. Let me know if you have any further questions.