How long will it take for an object to stop with a defined force over time?

[Samwell]

Homework Statement

How long it will take to stop the object?

Relevant Equations

v=∫F/m dt + v0
s=∫v dt + s0

Hello, I have the force defined as a function of time, where F=A-Bt and A=100N, B=100Ns^-1. I have to determine, how long it will take for object to stop, if t₀=0s and v₀=0,2ms^-1 and mass of the object is m=10kg. Can somebody please help me with this, because I'm having hard time with this task.

 

[Andrew Mason]

Hi Samwell. Welcome to PF!

What have you done so far? Have you tried plotting acceleration as a function of time? What would the area under such a graph represent? How is that related to solving your problem?

AM

 

[Samwell]

Hello, no, I have tried to solve the integral for velocity, but I wasn't successful. So when I know that F=ma and therefore a=F/m, I can solve the acceleration, but I'm not sure what should be the next.

Thank you.

 

[Andrew Mason]

Ok. So what is the condition for stopping (in terms of your equation for v)? What does that tell you about what the area under the graph of acceleration vs. time (the integral) has to be equal to?

Hint: To solve the integral, after substituting for F, break it up into the sum of two integrals.

AM

 

[Samwell]

So acceleration has to be negative and grow. But I still have no clue how to get that time from it.

Am I right with that?

 

[PeroK]

So acceleration has to be negative and grow. But I still have no clue how to get that time from it. View attachment 243320
Am I right with that?

Yes. That's right so far. (As as aside, note that by plugging in the numbers you have technically lost sight of the units, so you have a dimensionally unbalanced equation. Perhaps not worry too much about that.)

 

[Samwell]

But I still don't know how can I determine that time from my task.

 

[PeroK]

But I still don't know how can I determine that time from my task.

Have you ever heard of a thing called a quadratic equation?

 

[Samwell]

Yes, but I don't know how to apply this for that problem.

 

[PeroK]

Yes, but I don't know how to apply this for that problem.

It's difficult to see the problem. Normally with a quadratic equation the approach is to solve it. Quadratic formula? Completing the square?

 

[Samwell]

Okay, I've done it thank you very much. Just wondering which root I have to choose (actually I know which one because I have the result but I probably wouldn't if I didn't have). Can you please shortly explain why it has to be the second one in this case.

 

[PeroK]

Okay, I've done it thank you very much. Just wondering which root I have to choose (actually I know which one because I have the result but I probably wouldn't if I didn't have). Can you please shortly explain why it has to be the second one in this case.

One root is for a time earlier than $t_0$.

 

[Samwell]

Ok, thank you very much.