Question

If time t be regarded as a function of velocity v,prove that the rate of decrease of acceleration f is given by
F3d2t
Dv2

Answer 1

Given velocity 'v' and time 't' , find rate of decrease of acceleration 'f' .  

Explanation:  

• let the displacement be denoted by 'd'.
• let the acceleration be denoted by 'a'.
• given time is denoted by 't' and velocity is denoted by 'v'.
• Then we have,  the rate of decrease in acceleration 'f' as,                              $f=\frac{da}{dt}$    ---(a)
• now we know that velocity is the rate of change of velocity we get, $a=\frac{dv}{dt}$  ----(b)
• now substituting (b) in (a) we get,  $f=\frac{d^2v}{dt^2}$  
• here since 'f' is the rate of decrease in acceleration it will be considered negative mathematically.

Answer 2

Answer: The acceleration f is given by $f = \frac{d^2 v}{dt^2}$

Step-by-step explanation:

According to given question time is regarded as a function of velocity.

Hence acceleration, which is defined a change in velocity over time, will be written as:

Acceleration

$a = \frac{dv}{dt}$

It is also mentioned in the question that f is the rate of decrease of acceleration, hence f is written as:

$f = \frac{da}{dt}$

Replacing the value of a from above to this equation of f.

$f = \frac{d}{dt} (\frac{dv}{dt} )$

∴   $f = \frac{d^2 v}{dt^2}$

Hence proved.

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