Catapults have two problems:
You have only until the arm releases the capsule to accelerate the vessel. That means you have a fraction of a second to reach cruise speed. Such acceleration would turn your victims' insides into paste.
Friction would quickly deccelerate the capsule, so it wouldn't go very far. Supposing the capsule is as aerodynamic and heavy as a Cessna 172 as it reaches the highest point in its trajectory at the Cessna's cruise speed (226km/h, or 140mph), it will behave exactly the same as a Cessna 172 with its engines off.
A plane at cruise speed with its engines off does not drop as a stone. Rather, it behaves as a glider. It just so happens that planes that are not designed as proper gliders epically suck at gliding. You can play with Microsoft Flight Simulator or Kerbal Space Program for an approximated idea - just reach cruise speed and altitude with any self-powered aircraft, shut the engines down and try to land. It's actually possible to land a Cessna without too much damage, but if I ever had to go through that as a passenger in real life I would never fly again.
There is a way to solve 1 and reduce problem 2 if you are flexible. You wanted to use catapults, so we have been meddling in the realm of wacky engineering from the start. Let's up the ante and replace the the catapult assembly with guns.
Someone once asked Randall Munroe (a honorary god to worldbuilders) whether it would be possible to assemble a machine gun jetpack. Mr. Munroe always researches and answers scientifically, and his conclusion was that a jetpack would be unfeasible - but a machine gun powered plane would be a no brainer.
The A-10 Warthog has a machine gun (the GAU-8 Avenger) that produces 62.5% of the thrust of its twin engines, so it runs the risk of stalling when firing straight forward.
If you replaced the two engines with two other GAU-8 machine guns, it would be able to accelerate and fly faster!
But we can improve it further. Mr. Munroe says:
As good as this gun would be as a rocket pack engine, the Russians built one that would work even better. The Gryazev-Shipunov GSh-6-30 weighs half as much as the GAU-8 and has an even higher fire rate. Its thrust-to-weight ratio approaches 40, which means if you pointed one at the ground and fired, not only would it take off in a rapidly expanding spray of deadly metal fragments, but you would experience 40 gees of acceleration.
But if you somehow braced the human rider, made the craft strong enough to survive the acceleration, wrapped it in an aerodynamic shell, and made sure it was adequately cooled ...
… with a GSH-6-30, you could jump mountains.
Since your goal seems like giving your passengers a slightly worse experience than nowadays current airline flights, machine gun engines will at least make it survivable.