# Person at rest vs jumping on surface

I'm currently learning Newton's Laws, and I came across this question:

"A man lightly walks across a roof so that he does not break it. However, his friend tosses him a tool, causing him to jump in the air, and break through the roof. Why does the roof hold his weight when he walks, but not when he jumps?"

My confusion stems from the fact that in both cases there is a force of gravity and normal force acting upon him when he is on the ground. The only difference is his acceleration. And yet, if there is acceleration, there must be unbalanced forces, but what are those unbalanced forces caused by? How does he break through the roof?

You have already correctly identified the forces. The unbalanced force is simply the normal force being larger than before.

The normal force must now, apart from holding back against his weight, also create the upwards acceleration. Thus it grows.

If the surface is not strong enough to exert this new larger normal force, then the surface will break.

Ordinarily, when the man is standing or walking on the roof, his weight is acting downwards and is cancelled by a corresponding upward force from the roof, equal in magnitude with his weight.

If the man jumps in the air, his legs have to impart a force greater that his weight in order to accelerate his body, so the roof has to bear that extra downward force from his legs, which it might safely do.

When the man lands back on the roof after jumping in the air, the man's body must be suddenly brought to a stop, which will require a momentary force far in excess of his weight, and that is too much for the roof to bear, so the man falls through it.

You might find it helpful to consider how the roof responds to forces imposed on it. If the roof is supported by timber rafters, say, the rafters will sag somewhat under and imposed load. The sagging bends the wood and sets-up extra 'restorative' forces that resist further sagging, much as a spring under load compresses to the point at which the restorative force caused by the compression equals the applied load.

If a man stands on the roof, the rafters sag some more until the restorative forces within the timbers cancel out the man's weight, at which point they sag no further. If the man lands on the roof at some speed, the impact causes the beams to sag beyond the point at which the rafters will break, so the roof can no longer generate a restorative force to support the man.

I will treat the $$y$$-axis to be vertical, with the upwards direction being positive. When the man is walking across the roof, there are two forces acting on him as you say: the force from gravity downwards ($$-F_g$$) and the normal force from the roof upwards ($$N$$). The normal force is one half of an action-reaction pair, the other of which is the force from the man pushing down on the roof to keep him standing; Newton's third law says this is downwards with magnitude $$N$$.

As the man is not accelerating in the vertical direction ($$a_{\textrm{net},y} = 0$$), Newton's second law says there is no net force acting on him. This means the magnitude of the gravitational force and normal force must be equal: $$F_{\textrm{net}, y} = N - F_g = m_{\rm person}a_{\textrm{net},y} \implies N=F_g.$$

In order to jump upwards and catch the tool, the man has to be accelerating vertically ($$a_{\textrm{net},y} > 0$$). Netwon's second law then says that there has to be a net force in the upwards direction for this to happen, which then means that the magnitude of the normal force is greater than the force from gravity: $$a_{\textrm{net},y} > 0 \implies F_{\textrm{net},y} =N-F_g > 0 \implies N>F_g.$$

The net upwards force arises from the man jumping, where he pushes down on the roof in order to increase the normal force of the roof on him via Newton's third law, as mentioned above.

In the first case person imparts only $$\mathbf F_{total} = \mathbf W = m\mathbf g$$ on the roof. While in the second case total effect on roof is $$\mathbf F_{total} = \mathbf W + \frac {\delta p}{\delta t} ,$$ i.e. roof absorbs momentum change of a person. You can see it also as a version of conservation of person total energy and/or momentum. Because roof is weak in this particular case, and can't convert person kinetic energy to other forms of energy - heat, roof "deformation work", etc,- it just breaks up reducing person kinetic energy a bit, and then let it go down further. If person would be falling on say concrete, - outcome would be reversed, - concrete has abnormally high stiffness, so would absorb all kinetic energy of a person, make a heat a bit, and make "deformation work" on the falling person back.