Hello Dean and Linda:
>>? How? often will we have to stop on our 5 hour trip to pick up our RV to let our cat go to the bathroom--what's an average cat's range?
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If I understand you correctly, you plan to bring the kitty along with you in a car when you pick up the MH. I would strongly urge that you have someone take care of the cat until you can break her in properly to MH living. Cats do not like their environment to change, especially to living quarters that move and make noise.?

The car trip would be bad enough, but to then put her into another strange moving thing will drive her up the wall and then you will have an even tougher time getting used to the new environment.
>>? I've read that some people keep the litter pan in the shower.? How do you prop open the door so it doesn't bang and move as you travel? Our cat is a true "scaredy cat", and the shower door needs to not move at all or she'll never go in there.
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I've had several cats in my RV during 10 years of fulltiming plus have?rescued a few strays -- and never used the shower, tho some do and that is an OK place. My spot has always been in the bathroom on the floor somewhere. Others have put trap doors down into a compartment. Whatever you decide, you should have that liter box there (and her scratching post) "before" she enters the MH the 1st time. When you first put her in, put her right in the liter box. That way she will know exactly where it is. Also, be sure an have some of her used liter in the box.
>> If I stop at a rest stop and put her on a leash, will she relieve herself?
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I doubt it. Not unless that is the way she has always gone to potty. Again, my advice is to leave her somewhere else until you pick up the rig and decide exactly where to put the liter box. All of my cats were leash trained -- but none of them would go the bathroom anywhere but their liter box. And that's the way it should be.
>> Why do some of the RV parks? say "NO PETS"?? Why do they care if we have a cat?? Thanks for any tips you can give me.
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PETS is primarily referring to dogs. If not properly cared for, they can cause the park owner a lot of grief. Nothing worse than being parked next door to a barking dog left alone all day by their owners. Some do not pick up after pets, and so forth. Lotsa reasons. However, parks that do allow pets, and deal with the downside will get a lot more business. Many, many RVers have pets as part of their family.
Along with what I mention above about changes in a cats environment. The best way to get a cat used to a car or RV is to first put their liter box on board, "then" place the cat in it when they are first placed in the rig. Do not have the engine running at the time. Give them time to check everything out. She will enjoy that task.
Next, temporarily take her out of the rig, start the engine and let it warm up to an idle. "Then" bring her back in. Let her then get used to that new noise.
Finally, "very" slowly begin to drive the rig. Try to do this where there is little other traffic. This has worked well with every cat I've owned. It is very important because if she has a bad 1st experience, she may never recover. I hate to see cats hiding under couches and terrified when the rig starts.
If you do have to take her in a car for the rig pick up. Follow the same procedure before you leave in getting her used to the car, plus let ride in her carrier w/the door open occassionally. If she used a liter box, put it in the car or a smaller version of it after she is OK with the car motor and all.
Good luck -- hope it works out . . .