I used to think that when going around one of those curvy ramps I should stay on the outside. My reasoning was that the turn would be less sharp, and there would be less centrifugal force so I could still go fast without flipping over. However, I always wondered why turns felt smoother and better whenever I made really tight turns instead of wide turns.

After quick physics reasoning and a recall of a high school formula it becomes obvious that when turning one should stay on the INSIDE.

Centripetal force = (v^2) / r

Circle circumference = pi*2*r

velocity = distance / time

We are judging the viability of bigger vs smaller radius given that we complete a ramp/turn in the same amount of time. For example, if I go around the ramp choosing a wide turn in 20 seconds, vs going around the ramp choosing a tight turn in 20 seconds, which one puts less force on my car?

The tight turn has a smaller r, but also a smaller v. We know that as the r decreases, the v will decrease in a way linearly proportional to r because circumference is linearly proportional to radius, and velocity is linearly proportional to distance (given the same amount of time). But now as you can see from the top equation the v is SQUAREDLY proportional to my centripetal force whereas the r is only linearly proprtional to my centripetal force. Therefore even though the v and the r will decrease at the same rate as the circle gets tighter, that also means that the overall centripetal force is DECREASING as the circle gets tighter (because the v decreasing has a greater effect on the force's value than the r decreasing).

Therefore if you want to get somewhere quickly without flipping over your car the best way is to make a tight turn. Even though you can't go as fast on the turn, the smaller distance you have to cover more than makes up for it. If you take the same amount of time in boths situations, a tighter circle will put less force on your car. Similarly, if you keep constant the amount of force you put on your car in both situations, a tighter turn will be completed sooner.

If this proof is not well-written or needs clarification please let me know.

Good one and clear understandable logic. However, I believe in practical scenario... we also need to consider the time and effort taken to slowing to take a tight turn and then the time taken for accelerating again :)

ReplyDeleteIndeed, in the event that we are trying to maximize fuel economy, it might be better to just coast fast around the outside (but there are many variables involved). Thanks for your comment!

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