Question

The distance S an object travels under the influence of gravity in time t seconds is
given by S t( ) = + gt at +b 1
2
2 where, (g is the acceleration due to gravity), a, b are
constants. Check if the function S t( )is one-one

Answer 1

Answer:

Hey!❤️

Taking derivative δy=− 21

δgt 2−gtδt

But y is measured exactly ⇒δy=0

⇒8t=− 2sδgt

⇒ Tolerable percentage error in t is half that in g

⇒ percentage error in t= tδt

= 20.1%

= 0.05 %

Absolute error δt=(0.05%) x (1.43 s) = 0.7 ms

=7×10 −4s

Hope it will be helpful ✌️

Answer 2

$\huge{ { \pink{solve }}}$

It is given that, distance traveled (S),by an object travels under the influence of gravity in time t seconds is given by S(t)

${ { \blue{ =\frac{gt^2}{2}+at+b= }}}$

To ,Check whether the function,t→S(t),is Injective or not,

If For, Different Value of t,means at different instances of time,there are different value of S, also, there may be some S,which may remain unmatched,or there are no t's (time in seconds) for that S,then the function will be Injective.

Rule for one -one

If x, and y are two different instances,for two different values of t,then

If ,S(x)=S(y)

then, x=y

Applying the rule

$</p><p>\begin{lgathered}=\frac{gx^2}{2}+ax+b = \frac{gy^2}{2}+ay+b\\\\\frac{g(x^2-y^2)}{2}+a(x-y)=0\\\\(x-y)\times\frac{g(x+y)}{2}+a=0\\\\ x-y=0\\\\x=y\end{lgathered}$

Hence,

Proved The function,

${ { \green{ =\frac{gt^2}{2}+at+b= }}}$

is Injective(1 - 1 type of function).