Why is it harder to jump off a surface like a plank over water than on dry land, according to Newton's third law?

Question

Why is it harder to jump off a surface like a plank over water than on dry land, according to Newton's third law?

Via linear momentum or by the third law, I can't understand exactly

Answer 1

For understand conservation of momentum, it is necessary to make a sort of abstraction from everyday life experience.

For example, collisions in outer space, or at least in a frictionless surface, without air resistance.

In everyday life, momentum is conserved when we realize that the earth moves slighly backwards when we jump forwards. But because the earth movement is vanishing small, everything passes as we could change our linear momentum, while the ground is kept at rest.

The plank on the water is a situation in between that extremes. For any jump to a side there is a non-negligible movement of the plank to the other side because there is no static friction with the water. So, our net displacement depends on the mass of the plank. Only if it is big enough to have a small displacement, it is possible for us to jump close to which is possible on soil.

Answer 2

How high/far you can jump depends on how much force the surface under your feet pushes you with. Or, to be more precise, it depends on what momentum the surface imparts on your body( which depends on the force it applies on your body)

Now, the force your body experiences depends on the material too; see these two answers ([1], [2]) for the behavior of a sponge vs hard ground under force.

Imagine a surface (say the surface of a huge piece of sponge) that easily yields to the force you apply on it. As you make an effort to apply force on it (and as a result expect yourself to get pushed upwards, by Newton's third law), it experiences and exerts a force on you that is lesser than what would happen on hard ground. As a result, you would jump to a lesser height/distance.

Why does it depend on the material? Well this example might help you: suppose you jump onto a hard surface and a sponge from the same height. your intuition is likely to go wrong and tell you that you would be applying the same force on both of the surfaces as you land on their surfaces. However, you know that you experience lesser force from the sponge than from the ground-you feel it, and hence by third law you are also exerting lesser force on the sponge than on the ground though you jumped onto them from the same height. The sponge exerts a lower force because, force it exerts on you is actually proportional to the rate at which your momentum is changed- you know that the sponge compresses and takes more time to bring you to rest than the ground does, and hence you experience more force from the ground.

Similarly, your weight as measured on a weighing machine in an elevator accelerating downwards will be lesser than your actual weight because it measures the normal force, which is lesser than the normal force that stationary hard surfaces would exert.

In a similar way, when you push on a surface that yields/moves due to force, the force you experience is lesser than that you experience from hard ground and the same with the force you apply on them too, though it is hard to digest that you will actually be exerting a smaller force on the surface that yields, might have to study biology to understand why it is harder for your muscles to exert more force on a surface that yields :) Or, it might simply be that it is harder to apply a downward (contact-)force on anything that is already moving downwards.

You can in the same way argue in the case of a plank on water.