A friend and I are trying to figure out who is doing more work benchpressing? Me? Him? Or is it the same amount?

Question

Me and a friend have been working out for some time and have been bench pressing each week, increasing our weight. Once we started getting to heavier weight we noticed that he was able to get more reps of bench presses than I. We thought this was strange because we both know that I am stronger than he (only because I've worked out longer and can push heavier weight when it comes to other exercises, or at least this is what we think).

So I thought, well wait, can't I just calculate this out?

My friend, we will call him $A$, and we will call me $B$

So I know $F = ma$

So I know that the weight we are bench pressing is $215$lbs which is = $97.52$kg

So using this information I calculate the amount of force that is needed to push this weight from rest (off of our chest) to the highest point out arms will go.

$\therefore F = (92.52 kg)(9.8 m/s^2) = 955.696 N $

My friend, $A$ is 5'8 with a wingspan of about 6 foot which is = $1.8288 m$

I, known as $B$ am 6'1 and have a wingspan of about 6 foot 4 which is = $ 1.9304 m$

Knowing all this info I calculated that since

work = (force x distance) $W = FD$

so:

$W_A = (955.696 N * 1.8288m ) = 1747.77 J$

then the amount of work my friend would have to do to get 1 rep of a bench press movement would be $W_A = 1747.77 J$

so we also have: $W_B = (955.696 N * 1.9304m ) = 1844.87 J$

So the amount of work I would have to do would be $W_B = 1844.87$ so the difference between these two numbers is $5 percent$ so would it be safe to say that I have to do $5 percent$ more work than my friend to do one bench press movement because I have a longer distance to travel? I'm not sure if I calculated this right because I haven't done physics in a while, I was curious if the experts here at stack would know if this is a reasonable way to calculate this or if I have done so in error. Thanks in advance

Answer 1

The equations and idea in your calculation is correct. However, you used the wrong distance $D$. In a bench press, the weight does not move a distance of your wingspan, but rather the distance from your chest to however far your arms reach. It should be somewhat less than half of your wingspan.

So you need to measure that distance for both of $A$ and $B$, call them $D_A$ and $D_B$, and then recalculate $W_A=FD_A$ and $W_B=FD_B$. Then you'll have more accurate numbers to get a better estimate of the percentage.