Question

are all the speeds equal?

Answer 1

Since they are projected with the same velocity, that means they have the same kinetic energy. So, basically, when the three plums reach the same height, they should have the same potential energy. Now, since, the kinetic energy of a body is transformed into P.E. with height, so all three suffer same change in K.E. hence, their speeds should be equal.

Answer 2

If what @Sourav12084 says is right, then let's consider a simple problem. I throw a plum directly to the right and one directly up at the same speeds. After a certain time we look:

Plum that was thrown directly up has just reached the peak of its arc and has 0 speed. The other plum however never slowed down and in fact accelerated because the force of gravity pulled it down.

How is this much different than the case described where one plum is thrown straight up and one thrown at an angle? Are you sure?

Answer 3

Maybe this isn't that tricky and I'm just wasting your time, but hopefully not. It seems to be a contradiction but what I'm saying is actually different than what the question is talking about.

Answer 4

The one that you'll throw to your right will also reach a point where it'll have attained a max height and where its vertical component of velocity will reach zero....like the one you'll throw straight upwards

Answer 5

No, since when you throw it straight to the right there is never a max. It's at its max when it's thrown. See:

Answer 6

The point I'm making is that they have the same speed at that height of the dotted line. But they DO NOT reach the dotted line at the same time.

Answer 7

This situation is completely different from that asked in the question. you have already reached a maximum height. that means you have already given it a maximum potential energy at its start.
But in the question, all three plums start from zero potential energy.

Answer 8

It's not completely different, it's just highlighting a point. Because particle b's y-component of velocity is smaller than a's y-component it takes longer for it to get to that point. So they won't have the same speeds at the same time in general ever again.

They only have the same speeds at the same heights. The time it takes for them to get there is completely different though and ball b will hit the ground first while ball a will hit the ball last.

However WHEN they each individually hit the ground they will have the same speed. Does this make sense?