Usually what happens is that Time Machine prompts you for what to do, giving you two choices. One is to "inherit" the backups that are already on the drive, which will grant you full access to the backups and will cause Time Machine to continue incrementing off of them. Choosing to inherit the backups prevents the original computer from being able to use the backups, which in your case won't be an issue.
The second option is to leave the backups intact and start new backups on the drive. You won't be able to access those other backups and Time Machine won't "prune" them as disk space is used on the drive, but Time Machine won't wipe them, either. It does this under the expectation that the drive might be used by multiple computers for Time Machine backups.