Name: Inertial Frames of Reference
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Imagine if you are on a bus or a train, can you say you are in an inertial frame of reference? If yes, how can you predict it? Let us explore more in this vi...

Question

Prove that instantaneous acceleration is the slope of the velocity-time curve at the relevant point.

Accepted Answer

Instantaneous acceleration $a$ , or acceleration at a specific instant in time, is obtained using the same process discussed for instantaneous velocity. That is, we calculate the average velocity between two points in time separated by $∆ t$ and let $∆ t$ approach zero. The result is the derivative of the velocity function $v (t)$ , which is instantaneous acceleration and is expressed mathematically as

$a (t) = \frac{d}{d t} v (t)$

Thus, similar to velocity being the derivative of the position function, instantaneous acceleration is the derivative of the velocity function. We can show this graphically in the same way as instantaneous velocity. In figure below, instantaneous acceleration at time $t_{0}$ is the slope of the tangent line to the velocity-versus-time graph at time $t_{0}$ . We see that average acceleration $a = \frac{∆ v}{∆ t}$ approaches instantaneous acceleration as $∆ t$ approaches zero. Also in part (a) of the figure, we see that velocity has a maximum when its slope is zero. This time corresponds to the zero of the acceleration function. In part (b), instantaneous acceleration at the minimum velocity is shown, which is also zero, since the slope of the curve is zero there, too. Thus, for a given velocity function, the zeros of the acceleration function give either the minimum or the maximum velocity.

Prove that instantaneous acceleration is the slope of the velocity-time curve at the relevant point.

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