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

Show that the instantaneous acceleration at any point in the velocity-time graph equals the slope of the tangent to the graph at that point. Which one is more important in the laws of mechanics - the average acceleration or the instantaneous acceleration ?

Accepted Answer

The slope of a velocity graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.

The slope of a velocity graph will be given by the following formula:

$Slope = \frac{v_{2} - v_{2}}{t_{2} - t_{1}} = \frac{∆ v}{∆ t}$

Consider a velocity time graph as shown below.

Since $\frac{∆ v}{∆ t}$ is the definition of acceleration, the slope of a velocity graph must equal the acceleration of the object.

This means that when the slope is steep, the object will be changing velocity rapidly. When the slope is shallow, the object will not be changing its velocity as rapidly. This also means that if the slope is negative—directed downwards—the acceleration will be negative, and if the slope is positive—directed upwards—the acceleration will be positive.

The slope of the curve is positive between the times $t = 0 s$ and $t = 2 s$ . Since, the slope is directed upward, this means the acceleration is positive.

The slope of the curve is negative between $t = 2 s$ and $t = 8 s$ . Since, the slope is directed downward, this means the acceleration is negative.

At $t = 2 s$ , the slope is zero since the tangent line is horizontal. This means the acceleration is zero at that moment.

Average acceleration is the notified change in velocity for the whole journey. Whereas, the instantaneous acceleration is the acceleration due motion of the moving body at every instant. If the body is moving along a variable force field would feel different instantaneous acceleration with a different average one. Both are important depending upon situation and necessity.

Show that the instantaneous acceleration at any point in the velocity-time graph equals the slope of the tangent to the graph at that point. Which one is more important in the laws of mechanics - the average acceleration or the instantaneous acceleration ?

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