My vehicle needs to pace another vehicle

[charlievictor]

This is not really homework but it is work (fun) and I'm doing it at home. I’m a software guy and haven’t done a lot of physics in awhile. Actually, I’m building a dial display for a car rally computer. It’s basically a clock like display that reads out the odometer. I’m working on the stepper motor control loop for this. I’m trying to predict where the dial needs to move to in order to synch up with the car’s position. In order not to confuse the issue too much, I’ve boiled it down to the following problem. I think I have a solution that works except that the answer depends on the position where the two vehicles merge (t₃). Is there a way solve this without having to know t₃?

Homework Statement

• I’m in a vehicle traveling at a constant speed of v₀. I could be traveling forward (+) or reverse (-).
• Another vehicle is traveling at a constant speed of v₁. It also could be traveling forward (+) or backward (-).
• The other vehicle is a distance of d₁ from me at the start (with both vehicals moving). It could be ether in the positive or negative direction.
• I want to change my vehicle speed while undergoing a constant acceleration (a) and de-acceleration (-a) to arrive at the other vehicle and pace it at its speed of v₁.
• I would like to know what my maximum speed (v₂) I would reach during this maneuver.

Homework Equations

(I'll try to attached my scanned in PDF but here goes)

d₂ = 1/2at₂² + v₀t₂
v₂ = at₂ + v₀

d₃ = -1/2a(t₃-t₂)² + v₂(t₃-t₂) + d₂
v₃ = v₁ = -a(t₃-t₂) + v₂

d₃ = v₁t₃ + d₁

I know v₀, v₁, d₁, and a.

I want to solve for v₂ without depending on t₃.

The Attempt at a Solution

(see my attached my scanned in PDF)

v₂ = Sqr((2av₁t₃ + 2ad₁ + v₁² + v₀²)/2)

I plotted this in Excel and it seems to work.

But it seems to me that I should be able to sove this without using t₃ or d₃.

Thanks in advance.

 

[charlievictor]

I figured it out.

For the part of my problem where my vehicle is accelerating to meet the other vehicle, this equation applies:

t₂ = (v₂ - v₀) / a

For the second part where I'm trying to slow down to pace the other vehicle, this one applies:

t₃ - t₂ = (v₂ - v₁) / a

If I solve for t₃ I get:

t₃ = (2v₂ - v₁ - v₀) / a

I my attached PDF work notes, the last equation was this:

2v₂² = 2av₁t₃ + 2ad₁ + v₁² + v₀²

If I substitute in my t₃ from above, I get this:

0 = 2v₂² -4v₁v₂ + (2v₀v₁ + v₁² - v₀² - 2ad₁)

I used the quadratic formula to solve this for v₂.