Consider this :
First there was the 'sling' that was a string that threw a rock.
Then, they attached a stick to the string, and got more distance.
Then they tried bending the stick, and seeing if they could use the 'spring' of the stick to launch the projectile.
Then they got longer sticks. Then they tried tying down the bent stick (weapon loaded and ready).
Eventually someone tied the string to both ends of the bent over stick. During a struggle over it someone pulled the string back, and then let go. A 'lightbulb' went off in the other's head, and he started dreaming up what kind of projectile could be launched by this new found 'power'.
You know the rest.
I envision a vine crossing a path between two trees.
A wandering human passes by and trips over the vine that springs back to its original position.
The human investigates the cause of his fall and notices that the vine is capable of storing energy. He places his hand on the vine and pulls it back, then places a stick in front of it to hold it in place. When he lets go of the stick, the vine launches the stick several feet away.
After considering this effect, the human makes a portable version that can throw a stick farther and farther. Eventually, a crude but effective stick throwing device is produced.
But now the stick itself must be fine tuned in order to get more range, accuracy and killing force. Straighter sticks fly more distance than bent sticks. Certain types of wood materials produce better sticks. Eventually, through trial and error, a stick with a sharp point, hardened in hot coals is produced. At some point the stick is fletched with feathers to make it fly straight, and eventually the feathers are canted to make the stick spin and improve its accuracy.
From there improvements were made in materials and workmanship until a modern looking bow and arrow were made. The two units had to develop simultaneously with a definite goal in mind.......