Understanding Acceleration and Center of Mass in Shock Absorption

[ThEmptyTree]

Homework Statement

The compressive force per area necessary to break the tibia in the lower leg is about $F/A = 1.6 × 108 N/m^2$. The smallest cross sectional area of the tibia, about 3.2 cm2, is slightly above the ankle. Suppose a person of mass $m = 60 kg$ jumps to the ground from a height $h_0 = 2.0 m$ and absorbs the shock of hitting the ground by bending the knees. Assume that there is constant deceleration during the collision with the ground, and that the person lowers their center of mass by an amount $d = 1.0 cm$ from the time they hit the ground until they stop moving.

(a) What is the collision time $\Delta{t_{col}}$, to 2 significant figures?
(b) Find $N_{ave}$, the magnitude of the average force exerted by the ground on the person
during the collision in Newtons.
(c) What is the ratio of the average force of the ground on the person to the gravitational force on the person? Can we effectively ignore the gravitational force during the collision?
(d) Will the person break his ankle?

Relevant Equations

Newton's 2nd Law: $\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$

I don't attempt solving a problem until I fully understand it, conceptually.

After the hit (when maximum velocity is reached) the person starts losing momentum, having a constant upwards acceleration. The forces acting on the person are gravity and the normal to the ground.
$$N - mg = ma$$
$N>mg$ and that's why the person suffers the shock.
My question is, how does the person lower the center of mass of its body, if the acceleration of the center of mass is
$$\overrightarrow{A_{cm}}=\frac{\overrightarrow{F_{ext}}}{M}$$
If the acceleration is upwards, shouldn't (hypothetically) the center of mass go in the direction of the acceleration?

One explanation that came to my mind was the fact that if we consider the body a reference frame which accelerates upwards, then the fictitious force would be downwards, but it would only apply to this case.

I am very confused, can someone explain this to me please?

 

[PeroK]

What would happen if you applied equal and opposite forces at either end of an object?

Consider its acceleration and compression force.

 

[ThEmptyTree]

What would happen if you applied equal and opposite forces at either end of an object?

Consider its acceleration and compression force.

I don't quite get it. I try to compare it to applying forces at both ends of a rope but I'm bad at visualizing things. I have never worked with compression forces and it hasn't been introduced in the course yet. Aren't the only external forces gravity and normal?

However, introducing the idea of compressing force might explain why the person has acceleration despite not moving, like an object being compressed. In this case the compression force is determined by the person crouching (so it's from the top)?

I have been meditating on what you said for half an hour, but no results. Further hints will be appreciated. Thanks a lot.

 

[PeroK]

Newton's second law describes motion based on the net external force, not total compression forces.

 

[haruspex]

If the acceleration is upwards, shouldn't (hypothetically) the center of mass go in the direction of the acceleration?

Don't confuse direction of motion with direction of acceleration. If you throw a stone up, even while it is rising the acceleration is downwards.

 

[ThEmptyTree]

Don't confuse direction of motion with direction of acceleration. If you throw a stone up, even while it is rising the acceleration is downwards.

Yup, I think that's it. The speed is still downwards. Thanks.