How can I determine if a certain machine exists on the network?
If I ping the IP address, would that be sufficient?
No - a machine is not obliged to respond to a ping. To be certain, you need to ping from the local network the machine is on, then check for an arp response:
If you see a mac address against the IP address, then the device exists. If not, then it is effectively not present (right now - it could just be powered off for the moment).
Pinging is not enough. Many firewalls will block pings.
You could try nmap
That attempts to discover host even if they are blocking ping scans.