The official vivid package for mongodb is 2.x, and the packages that MongoDB provide themselves on http://repo.mongodb.org/apt/ubuntu/dists/trusty/ are only available for LTS releases.
An alternative is that I could switch back to 14.04 LTS, but I would need to install openjdk-8-jdk, which is only available for vivid.
Best Answer
It's possible, that MongoDB doesn't start if you use the Trusty packages:
In this case, install the Debian Wheezy packages as described at the end.
Install the Trusty DEB packages
Download the DEB files
And install with (in this order)
And taken from the comments, thank you @Pilot6:
Use the Trusty PPA
Use the Debian Wheezy packages
It's possible, that MongoDB doesn't start if you use the Trusty packages:
In this case, install the Debian Wheezy packages as described below