I suspect that, less than an absolute "solution" or match, the usefulness was in limiting U-boats' tactical options:

• deck guns. Early ww2 U-boats had guns and liked to use them - saved on torpedoes among other things.

• surface, rather than submerged stalking and attacking. U-boats, until the Schnorkels came along (too late to do much), ran on batteries when at periscope depth. On the surface their diesels had better speed and endurance - 17kn vs 7kn for a TypeVII - and could better engage cargo ships.

Armed merchantmen would make both tactics risky, though their effect would be much less significant in a convoy situation with dedicated anti-sub military ships or ASW aicraft. And - at least in the case of just mounting a few old guns on a normal cargo ship, with military gunners (note that depth charges are not mentioned) - probably did not cost much to set up or divert many resources from other uses, especially in the early stages of the Battle of the Atlantic when the U-boats were the most dangerous and the UK had few other areas of direct combat to attend to.

As far as depth charges go, I think you'd run against another consideration. At the end of the day, while a gun might make sense for the reasons noted above, it does not undermine the cargo ship's basic role to get away and deliver its cargo. Dropping depth charges requires both specialized equipment and crew, but also requires a cargo ship to close in and fight with the sub, which does put its cargo at risk (destroyers getting sunk by subs was not uncommon and they are made for ASW). So that idea makes a lot less sense from a strategic perspective, beyond the exact tactical match up implied by the weapon systems.

To conclude: adding a few guns to a lot of transport ships - effective at dissuading subs surfacing, at least during the day, not much cost. Deploying full-on armed transport ships, like Jervis Bay, to aggressively "take down" subs or wage combat - not so much.

Armed merchantmen could clearly deal with a submarine attacking by day, on surface.

During the night, U-Boats surface attacks would include many submarines, and the capacity of numerous merchantmen to defend a convoy would clearly be about the tactical plot:

• Who would detect first?
• Detect before the launch of torpedoes?
• Meteorological conditions

And in the end, there are underwater attacks: the merchantmen were searching for the periscopes. But they were not equipped, nor had qualified personnal, to use passive or active sonar (ASDIC). This is neither the sort of boats that has the possibility to launch submarine grenades: they did not have room to embark rails, and mortars launching grenades were quite rare during WW2.

So this was a fast presentation on what were the means. To answer your specific question, you can say that merchantmen were useful to protect convoys from submarines, they had this role, but they were not a tool to defeat a submarine.

While armed merchantmen weren't particularly effective at actually fighting U-Boats, the mere act of arming them had a tactical and strategic effect. The Battle of the Atlantic could be won by hampering U-Boat attacks as much as sinking U-Boats.

U-Boats are not really submarines, they're submersibles; they spend most of the time on the surface. Underwater they're slow and half blind with limited air and battery life. They prefer to attack on the surface, especially earlier in the war before Allied air power became overwhelming. They're faster on the surface, and it saves the battery charge. Torpedoes are very expensive, and they carry a limited supply, so the deck gun is preferred to maximize their patrol time.

On the surface they can post multiple lookouts on top of the sail giving them more eyes and the higher position means they can see further. Submerged it's just one person occasionally looking through the periscope at a very low vantage point.

U-Boats are small and fragile. Any hit might disable their ability to dive rendering them a sitting duck on the journey home. Even an old 3" gun mounted on the back of a merchantman was a threat.

If a U-Boat captain thinks the merchantman is armed they may be more cautious. They may dive and approach underwater. This drains their batteries leaving them with less time for action. They're very slow and can't see as well. A lone, and presumably fast, merchantman might dodge or outrun a submerged submarine.

They have to expend a precious torpedo, rather than cheaper and more numerous deck gun ammunition. Even if the merchantman is sunk by that torpedo, its armaments made that kill more costly to the U-Boat. It will have to sail home for resupply earlier, or risk a surface action later on for a lack of torpedoes. At the scale and desperation of the early Battle of the Atlantic, such Branninganian tactics, literally make your enemy run out of ammunition, were a win for the Allies.

Arming merchants had a severe downside for the merchantmen themselves. It made them, in the eyes of the attacking U-Boats, combatants.

In both wars, particularly WWI, the U-Boat campaign began by respecting the cruiser rules. Unarmed civilian merchants were to be stopped, searched for contraband if neutral, and the crew allowed time to take to lifeboats before sinking. This obviously put the U-Boat at risk, but it was the agreed upon rules of war. To a sailor, the sea is more the enemy than the enemy.

To not follow the internationally agreed upon prize laws was to be branded pirates. This is one of the justifications used by the US to attack U-Boats while officially neutral.

The British began arming their merchantmen with old naval guns. Rather than having untrained civilians messing about, the guns were crewed by Royal Navy and Royal Artillery gunners. The rest of the Allies soon followed.

Germany complained they were no longer armed civilian vessels subject to prize laws, they were combat vessels crewed by soldiers. This is one of the justifications for unrestricted submarine warfare: approaching a ship to stop and search it was simply too dangerous for a U-Boat.

Q-Ships, while technically not a violation of the cruiser rules, made the situation worse. These were merchant ships outfitted with hidden guns and sailing alone to act as traps. When the U-Boat approached to investigate they'd wait until it was at point blank range, unmask the guns, fly up the naval ensign, and open fire.

Such false flag ruses at sea were a common and accepted part of naval warfare; so long as the ship raised the proper flag before opening fire it was considered a clever and just tactic. However, unlike a proper warship a U-Boat is extremely vulnerable in a surface action against even an unarmored merchantman. Q-Ships exploited the cruiser rules in a way that left U-Boat captains cautious about approaching any seemingly unarmed merchant.

Q-Ships, while not particularly successful at sinking U-Boats, did lead to the unraveling of the cruiser rules, and were another justification for unrestricted submarine warfare.

Question:
Were armed merchantmen effective against submarines?

They were called Q-Ships short for Queen Ships armed Merchantmen which convoyed with the merchant fleets for protection. No they weren't very effective.

The German's had 13, British employed 9, the United States beginning in January 20, 1942 outfitted 5 in the Atlantic and 1 in the Pacific, but ended their use in 1943 because they were "entirely unsuccessful".

Q-ships The careers of all five ships were almost entirely unsuccessful and very short, with USS Atik sunk on its first patrol; all Q-ships patrols ended in 1943.

Japan deployed one which was sunk by an American submarine on it's first voyage in 15 January 1944.

Q-ships (all British) accounted for about 7% of German uboats lost to enemy action during WWII. Compare that to 17% u-boat sinkings from depth charges and 20% from mines. Or compare all of Q-boat victories over uboats in the Atlantic in WWII (13), to the record of a single destroyer over just 12 days in the Pacific. The USS England which sank six Japaneses submarines over 12 days, which is still a record.

Stats on German Uboats Lost to Enemy Action

U-Boats
Of the 373 German submarines that had been built, 178 were lost by enemy action. Of these 41 were sunk by mines, 30 by depth charges and 13 by Q-ships.

No merchantmen were ineffective against submarines. If on surface the merchantmen were huge lumbering targets while the submarines were small with only the conning tower showing above the surface. If attacking submerged the submarine could torpedo a merchantmen from outside of effective range of a merchantmen's deck gun and be gone before the armed merchantman was ever able to fire. They only way the Q-boats really had a chance was if the submarine commander was out of torpedo's or was conserving them and surfaced close to the convoy.

The early strategy in the battle of the North Atlantic was to outproduce Germany. But early in May of 1941 the Uboats were sinking allied merchant men (the most dangerous service in WWII) by a ratio of 2 sunk for every 1 produced by British and American shipyards.

Merchant Men Production

Merchant Ships
During the Battle of the Atlantic in the early years of World War II, the German U-Boats were sinking allied merchant ships at an alarming rate. Unless more ships were built to replace those sunk, the war effort would grind to a halt. Cargo ships would have to be replaced faster than the U-Boats could sink them. In May 1941, the rate of German sinkings of merchant ships was more than three times the capacity of British shipyards to replace them, and more than twice the rate of combined British and American shipyard output at the time.

One of the most effective way to defeat the German tonnage-war was to build more ships than the Germans could sink.

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U.S. Merchant Ships Sunk or Damaged in World War II
According to the War Shipping Administration, the U.S. Merchant Marine suffered the highest rate of casualties of any service in World War II.

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Allied Merchant Ships Lost

The Battle of the Atlantic
The outcome of the battle was a strategic victory for the Allies—the German blockade failed—but at great cost: 3,500 merchant ships and 175 warships were sunk in the Atlantic for the loss of 783 U-boats..... Of the U-boats, 519 were sunk by British, Canadian, or other allied forces, while 175 were destroyed by American forces; 15 were destroyed by Soviets and 73 were scuttled by their crews before the end of the war for various causes.

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Uboats
By the end of the war, almost 3,000 Allied ships (175 warships; 2,825 merchant ships) were sunk by U-boat torpedoes.

another link for Q-Ships