Just a check list on whats going on in the image of the circuit on the bread board.
The Relay in image shows 5V DC, then a resistor for the transistor using a value of 300R (Ohm) would place 13mA current on the Transistors Base,
the Transistor collector connected to the relays coil should sink the relays coil current/Volt requirement of the Unknown relay current value,
5 Volt types are generally not too high in current tho some can be. Data on the relay would help so I am acquiring that. Then the other end of the coil to + power supply,
:notes: Relay coil does not have an polarity requirement.
Relay Data for that series, included in PDF form.
The coil shows 70R resistance at 5V so these are not like the panasonic works type that I initially referred them to, that is my mistake!
The relay shows to consume a decent 71mA at 5V, the 300R resistor on the unknown sinking Transistors Base is adequate.
What is the Transistors ID? the one connected to the relay.
I am just wandering of the Transistors current capability.
The motors current requirement is still unknown.
Also what is the motors power in Volts requirement if known?
And the resistor for the Capacitor timing part of the 555 timers discharge pin 7, resistor R1 with an value shown in image of 51K Ohm, if the circuit is functional to an point, try to increase that value to an higher level and observe when run if the motor operates longer as a result,
What name is that 555 Timer? A Manufacturer. those pins look Texas instruments.
And also in the relay datasheet it shows some info, the relay image shows normally closed type, not open.
That means the relay could be doing opposite of what you want to happen when operating the motor.
= relay off, motor runs. Relay on, motor stops.
If I'm seeing the datasheet correctly. As it is showing a single pole double throw SPDT type.
The issue may be with the relay. However if that relay you have shows pins similar to the datasheets pins, then the relay can be used, it just needs to have the motor wired to the other relay contactor pin.
Now I am not fully aware of that relay completely, however referring to the image provided, the upper violet (purple) wire can be brought over to the pin at the to left hand side, then the motor would be powered when the relay is ON. If it is a normally closed type.
The violet (purple) wire to the relays other contactor pin via the image, avoiding the Transistors Collector pin that would cause the Transistors collector to sink all available supply current that can exceed the transistors current rating and be damaged.