trouble w/ new jet pump system

Aloha!

I've just replaced an old jet pump for our water catchment system. Basically, my problem is that the motor won't turn off. If the water line is open and in use, the pump runs fine and pressure is fantastic. Once water is shut off, however, the pump doesn't just build system pressure and stop -- it cycles on/off very fast (like a stutter) until the motor heats up and cuts off. Then I need to unplug the pump, wait about 10 minutes and then I can plug it in to get it running. And repeat. If I increase the cut off pressure to avoid the stuttering, then the pump runs continuously.

I'll do my best to answer the following questions. Thank you in advance for your time!! This is my first pump installation and troubleshooting this has been a pain so any help is deeply appreciated. Here's a link to video of this: http://www.youtube.com/watch?v=e6tiL4CmG18&feature=youtu.be

Type of pump?
Submersible________
Two wire (no control)________
Three wire (control box)______
Wire Size_________ Wire Length________
or
Jet Pump (above ground)_____YES____
One or two pipes down the well__1__

Size of Pump?
Motor Horsepower?____3/4______
Pump Model #______DAYTON 4HEY5________
Date Pump Installed_____1/15/2014_____

Pumping from?
Cistern tank_____CATCHMENT TANK 10' TALL x 16' DIAMETER______
Pond, lake, river________
Water Well____________
Depth of well__________
Depth to water___7'______
Pump Setting__________
Pipe Size____1.25" INLET, 1" OUTLET_____
Drop Pipe Material
PVC___yes_____
Steel_______
Poly________

Well Recovery Rate_______gpm
Well Casing Diameter_______â€
Rock Well__________ Sand Well__________ Other______________
Date Well Drilled____________

Well Casing Material
PVC________ Steel_________ Other_________


Pressure Tank?
Bladder or diaphragm tank (one pipe to tank)____DIAPHRAGM - FAULTY: HUGE LEAK IN VALVE______
Size or model of tank____________
Air charge in top of tank, with pump off and water drained____________PSI
(check with car tire gauge)
or
Plain Hydro Pneumatic tank (two pipes to tank, one in and one out)_________
Size of tank________________

Pressure Switch Setting?
On 30, off 50 ___yes_____
On 40, off 60_________
Other_______________

Pump Control Method?
Cycle Stop Valve model #_________
Variable speed control #__________
Pump Start Relay (sprinkler timer, no tank)__________
Manually turned on and off____________

Pump Protection
Cycle Sensor_________
Pumptec_____________
Low pressure cutoff switch (lever on side)__________
Other_______________

Filters or Softeners______FILTER________
Before or after pressure tank___BEFORE____
Type of filter_________20 micron__________
Bypass available______NO bypass of filter; YES bypass to pressure tank__________

Water Used For?
House Use___YES____ Number of baths____2___ Number of People____3____
High Flow Showers_______gpm?
Plus/Or
Irrigation with timers____NO____
Irrigation with hoses____YES____
Heat Pump______gpm?


Problems Experienced
No Water_________________
Water only part time________
Water at all times but weak_____
Air in water_______________
Pressure surging___________
Water Hammer (noise)______THIS SOUNDS LIKE AN APPROPRIATE DESCRIPTOR
Too Much pressure_________
Other____________________


Pump makes clicking or buzzing sounds___ CLICKING COMES FROM THE PRESSURE SWITCH TURNING ON/OFF RAPIDLY
No Sounds______________
Pressure gauge reading____60____psi
Other____________________________________

Do you have, and know how to use
an Ampmeter and Voltmeter______YES

Describe Problem__________________________

How far is the pressure switch from the tank?

About 5'. I failed to mention that I've been bypassing the tank on account of its leak. This is a rookie question: do I need to have a pressure tank in order for the pump to function properly? I figured it would just turn on as needed to pressurize the system and then turn off.

Ya gotta have a tank and the pressure switch should be closer than 5' away