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A car starts moving with uniform acceleration $a$ and retards uniformly at uniform deceleration $b$ .Find the average speed of the car.

Total time = $t$

#Mechanics

Easy Math Editor

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Comments

Plot a velocity time graph with two lines : one with slope $+a$ and another with slope $(-b)$ .Then, average speed is given by the area under that graph divided by the total time interval, which can be easily obtained to be $\frac {abt}{2 (a+b)}$

I guess this is the simplest way.

Venkata Karthik Bandaru - 5 years, 10 months ago

Thank you very much.

Swapnil Das - 5 years, 10 months ago

An amazing approach which you can use to solve problems of thist type and many other in kinematics is applying time reversal. For example suppose we make a video of the body which initially accelerates with acc. a and decelerates with b. Now when you will see the video backwards the body initially acc. with acc b and decelerates with a. And the time taken for both the acc and deceleration will be same as before. (If it took 1sec to acc. initially then it would take 1 sec again when decelerating in the reverse video.)

Now continuing with the problem. Suppose acc. time is t(1) and decelarating time is t (2). Then velocity attained when the particle changes its acc. is

V=a.t(1)

and from reverse video method as described earlier it is the same v in opposite direction. V=b.t(2) So we get

at(1)=bt(2) and t(1)+t(2)= T

$t_{1}=\frac {b}{a}t_{2}$ $(1+ b/a)t_{2}=T$ Now we can determine t1 and t2 very easily. Average velocity is total distance / total time. Determine distance by using second equation of motion for the two time intervals t1 and t2 separately. The answer should come out to be $\frac {abT}{2 (a+b)}$

The approach is more useful in many other questions.

Satvik Choudhary - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$