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
nmap -PN 192.168.1.0
That attempts to discover host even if they are blocking ping scans.