I’m trying to get through 11-6 section of Feynman’s Lectures on Physics, Vol 1, particularly explanation of acceleration vector calculation in his example:

example figures

example description

It’s clear that acceleration vector equals Δv/Δt, but calculation of its components (tangent and transverse) is obscure to me.

Maybe I’m missing some elementary math or some insight behind working with limits, but:

  1. Why tangent acceleration is dv/dt?
  2. How it even possible to multiply a vector by an angle (v•Δθ)?
  3. Do I correctly understand that Δθ/Δt gives us an angle velocity and transverse acceleration is calculated as product of linear velocity times angular velocity (v•(Δθ/Δt))?

Could someone please explain that?

  1. An acceleration orthogonal to the velocity does not change the speed. For example an object on a circular trajectory with constant speed has only orthogonal acceleration. Therefore the rate of change of speed is the acceleration in tangential direction.

  2. $v$ is not a vector, it is the speed. Even if it where a vector, it would just be a multiplication of a vector with some scalar. Remember that an angle in radians is defined by $\theta = s/r$, where $s$ is the arc length of a circle and $r$ the radius. So an angle is just a dimensionless number.

  3. Technically yes, but thats not the point. $\Delta v_\perp = v \Delta \theta$ is the same equation as in 2, where $\Delta \theta$ is the angle, $\Delta v_\perp$ the arc length and $v$ the radius of a circle. Of course $\Delta v_\perp$ is a straight line and not a part of a curved circle, but in the limit of infinitesimally small $\Delta \theta$ the difference disappears, because for a very small angle the curved arc of a circle looks like a straight line.

  • $\begingroup$ Thank you, sir. I've found the example to be pretty confusing, as 2 different speeds are used and r1, r2 are used to describe distance from O, but approximated circle with radius R is used to describe the solution. I would decouple images for better clarity. For example, this image describes tangential acceleration separately pretty well and intuitively obvious: bit.ly/Centripetal-Acceleration $\endgroup$ – oblalex Sep 18 '19 at 21:11

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