MadSci Network: Physics Query:

### Re: what is the physics involved in a 400-meter hurdle race

Date: Mon Dec 2 13:12:01 2002
Posted By: Tom Cull, Senior Staff Scientist
Area of science: Physics
ID: 1038420438.Ph
Message:

Hi Vince,

Hi Vince,

Your question, “what is the physics involved in a 400-meter hurdle race?” is extremely open-ended.  But I will try to hit some highlights to help you along with your report.  In fact, the biomechanics and physics of running is a very rich field.   In addition, because the race involves hurdling, jumping has to be considered.

I did find an article entitled “Physics of Sprinting” by Igor Alexandrov and Philip Lucht in one of my favorite books (I often cite it), The Physics of Sports edited by Angelo Armenti, Jr.  There are several other articles on running and jumping.  Unfortunately, this book is difficult to find and much of the physics formulation is beyond 7- 9th grade level.

But anyway, the basic premise is that running, or at least sprinting can be modeled as a resistive force proportional to the speed and mass of the runner, and a propulsive force proportional to the mass and the force per unit mass produced by the athlete.

The equation of motion can be expressed as the formula:

d2 x(t)/dt2 = dv/dt = f – R * v (t),

where f is the force per unit mass of the runner, R is a constant of resistance, and v(t) is the speed at a function of time.  It turns out in this approximation, the equation of motion of the runner is the same as a particle dropping through a viscous fluid.

The other aspects to consider are the forces and torque generated by the body to produce locomotion.  You might want to focus on just the legs to model how much force and energy is involved in running.  Most teachers and scientists appreciate a reasonable approximation to the real system as a first step.

For example, You could try to figure how much torque generated by the hip and leg system is minimally required to move a lower leg.   This torque must be generated by the contraction of the quadriceps muscles (thigh muscles).

To figure out the torque we need to make approximations or guessed measurements based on an average person.  Let’s say that the leg bends as the quadriceps contract so that the system can be modeled as the weight of the upper leg located half way along the upper leg bone (femur) and that the lower leg weight hangs straight down at the knee.

Let’s solve the force and torque equation for the leg bend at 90 degrees at the hip and 90 degrees at the knee.  If the leg is stationary, then torques must balance.  Therefore, the torque provided by the quadriceps muscles attached between the hip and leg must balance the torque from the weight of the upper leg located at the center of the upper leg length plus the weight of the lower leg located at the length of the upper leg.

I hope you are still following my model.

This can be expressed as Sum of Torques = 0.

Torquehip-quadriceps-leg + Torqueupper leg + Torquelower leg = 0.

Torquehip-quadriceps-leg  - Mupper leg* Lupper leg/2 * g – Mlower leg * Lupper leg * g = 0,

where Lupper leg is the length of the upper leg bone, Mupper leg is the mass of the upper leg, Mlower leg is the mass of the lower leg, g is the acceleration of gravity.

This can be reduced to:

Torquehip-quadriceps-leg = Lupper leg * g  * (Mupper leg/2 + Mlower leg).

Let’s solve this equation assuming that our runner is 1 meter tall and weighs in around 76 kilograms.  Let’s further assume that she is half legs in weight and height.  So her legs  are 50 cm long and together way 38 kg.  Just for kicks (pun intended), we will say that the upper leg is 25 cm long and comprises 25 kg of the mass.  Which leaves 13 kg for the lower leg.  Further, we assume that the center of mass of the upper leg is half way, or in other words, 12.5 cm.

Torquehip-quadriceps-leg = 0.25 m * 9.8 m/s2 * (25/2 + 13) kg

Torquehip-quadriceps-leg = 62.5 N * m.

For this, we can make assumption about where the quadriceps muscles attach to figure out the force generated by the contracted quadriceps.  For simplicity, let’s say the quadriceps are effectively  the same length as the upper leg (it attaches just above the knee and just above upper leg socket) when contracted.

Torquehip-quadriceps-leg = Fquadriceps * Lupper leg

So,

Fquadriceps = Torquehip-quadriceps- leg / Lupper leg

Fquadriceps = 62.5 N * m / 0.25 m

Fquadriceps = 250 N.

This actually seems pretty reasonable.

If this is the level of physics detail you want to pursue you could continue this model to figure out energy required or power consumed during running.

There are few  previous related answers available on the Mad Scientist Network.

(some by me, so I apologize for the self- referencing):

Re: Would we run faster if the wind helped us. It can deley us if it´s coming a

Re: Why are bigger people harder to tackle

Re: Is it ever faster to dive instead of run to an object (i.e. first base)?

Re: What physic's are involved in trampolining

Re : VRD: Biochemical energy needed to jump a physical distance?

As a related aside you could do a web search on animal locomotion.  I know there has been much study of how dinosaurs might have walked or their top speed when sprinting.

Sincerely,

Tom “Sprinting Back from Retirement” Cull

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