Tough Physics integration problem dealing with kinematics.

[lorenzosounds]

Homework Statement

A train traveling at v0 = 22.0 m/s begins to brake by applying a velocity-dependent instantaneous acceleration a(v) = α /(v + u) m/s2, where α = – 23.0 m2/s3, v is the instantaneous velocity of the train, and u = 0.5 m/s. Determine the distance traveled by the train until it comes to a complete stop.

I know I need to integrate, but I don't know how to get to d(t).

Homework Equations

(v+u)dv = (α)dt

The Attempt at a Solution

-22v-11 = 23.0(dt) is where I'm at. I'm not sure how to integrate both sides. How do I get to v(t), so I can solve d(t)?

 

[TSny]

2. Homework Equations [/b]
(v+u)dv = (α)dt

Hello, lorenzosounds. You'll need to eliminate time, t, in favor of distance, x. Can you express dt in terms of v and dx?

 

[lorenzosounds]

Hello, lorenzosounds. You'll need to eliminate time, t, in favor of distance, x. Can you express dt in terms of v and dx?

Would it be -(1/2v^2+0.5v)=-23.0t ?

 

[TSny]

Would it be -(1/2v^2+0.5v)=-23.0t ?

No. If you integrate (v+u)dv = (α)dt, then the left side would be integrated from v_o to v. So, v_o as well as v would appear on the left side.

But, then you'll have to solve this for v as a function of t, which will get messy. After that you would have to integrate one more time to find the distance traveled.

There's an easier way. Instead of trying to get v as a function of t, try to get v as a function of x. Go back to (v+u)dv = (α)dt. Is there a way to express the differential dt in terms of dx? Hint: v = dx/dt.

 

[Nickie]

I understand that this is a challenging problem and requires a strong understanding of kinematics and calculus. To solve this problem, you will need to use the equation (v+u)dv = αdt and integrate both sides with respect to time. This will give you the equation v(t) = (-α/ln(v+u)) + C, where C is the constant of integration. To solve for C, you can use the initial condition that the train's velocity is 22.0 m/s at t=0. This will give you the value of C as -22.0 m/s.

Once you have the equation for v(t), you can use the definition of acceleration, a(t) = dv/dt, to find the expression for acceleration as a function of time. This can be integrated to find the equation for the distance traveled, d(t). Finally, you can use the fact that the train comes to a complete stop when its velocity is 0 m/s to solve for the time it takes for the train to stop. You can then plug this value into the equation for d(t) to find the distance traveled by the train until it comes to a complete stop. I recommend breaking down the problem into smaller steps and using the appropriate equations and initial conditions to solve for each step.

I hope this helps and good luck with your problem! Remember, in science, it's important to break down complex problems into smaller, manageable steps and use the appropriate equations and principles to solve them.