Suppose you catch a baseball, and someone invites you to catch a bowling ball with either the same momentum or the same kinetic energy as the baseball. Which would you choose? Why?
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To a certain extent, it does not matter which answer you provide. The phrasing of the question makes the answer to it subjective. You could say that you prefer momentum simply because you think it is more fun to experiment with; however, I suspect that your teacher would like a more scientific response. Your best bet is to plug in some hypothetical numbers in order to provide a more data-driven response.
The problem states that you caught a baseball that was thrown to you, but it doesn't state the speed of the throw. If the throw was slow enough, the differences between the momentum and kinetic energy answers will be negligible. For the sake of the argument, let's use a fairly standard throwing speed of 22 m/s. Momentum is calculated by multiplying mass (kg) and velocity (m/s). The standard mass of a baseball is .145kg.
.145 x 22 = 3.19.
Now let's calculate that same throw's kinetic energy. The formula is .5 x mass x velocity^2.
.5 x .145 x 22^2 = 35 joules
The next thing to do is figure out how fast a hypothetical bowling ball would be thrown at you to get the same momentum or energy.
A standard bowling ball is around 5 kg. Plug that number into the formulas and solve for velocity.
5v = 3.19
v = .638 m/s
and
.5(5)(v^2) = 35
v = 3.74 m/s
I would personally rather catch the momentum option because it is moving slower. Of course you could argue that you would rather catch the kinetic energy option because it is more fun catching faster throws.
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