A Projectile problem -- Time to reach 1/3 of the max height

[rudransh verma]

Homework Statement

A particle is projected vertically upward and it attains max height H. If the ratio of times to attain height h(h<H) is 1/3, find h.

Relevant Equations

$s=ut+\frac12at^2$

Assuming it’s one body whose initial speed is u. First it attains height h then H. t1 and t2 are two times at which they attain h and H.
$h=ut1-\frac12gt1^2$
$H=ut2-\frac12gt2^2$
$\frac {t1}{t2}=1/3$ Replacing t2 with 3t1, I am stuck.

 

[haruspex]

You have not used a certain fact. Can you spot it?

 

[BvU]

Make a drawing of height as a function of time. Is the question clear now ?
I mean iss the mathematical problem clear now ?

What is the relationship between $H$ and $u$ ?

[[edit] ah, haru was faster !

$\$

 

[rudransh verma]

What is the relationship between H and u ?

Maybe we can use $u^2=2gH$

 

[rudransh verma]

@BvU I got based on $h=ut1-\frac12gt1^2$ , $H=ut2-\frac12gt2^2$ , $u^2=2gH$ and $t2=3t1$
$\frac{u^2}{2g}=3ut1-\frac92gt1^2$ and
$h=ut1-\frac12gt1^2$
But then ?

 

[BvU]

Perhaps it's time to invoke another relevant SUVAT equation ... how about $u$ and the vertical velocity at $H$ -- what is their relationship ?

$\$

 

[haruspex]

@BvU I got based on $h=ut1-\frac12gt1^2$ , $H=ut2-\frac12gt2^2$ , $u^2=2gH$ and $t2=3t1$
$\frac{u^2}{2g}=3ut1-\frac92gt1^2$ and
$h=ut1-\frac12gt1^2$
But then ?

When manipulating simultaneous equations, you need to keep in mind which variables should appear in the final equation. The given variable is H, so don't try to eliminate it. The ones to be eliminated are t1, t2 and u.

 

[rudransh verma]

Perhaps it's time to invoke another relevant SUVAT equation ... how about $u$ and the vertical velocity at $H$ -- what is their relationship ?

$\$

I am getting $h=(5/9)H$ but it’s not the same in book.

 

[haruspex]

I am getting $h=(5/9)H$ but it’s not the same in book.

I get the same.
The easiest way is to reverse it and consider an object in free fall from rest. If it falls distance y in time t, how far does it fall in time 2t? What about time 2t/3?

What does the book get?

 

[rudransh verma]

What does the book get?

(3/4)H

 

[haruspex]

(3/4)H

That would be the answer if the time were a half, not a third.
Did you try the way I mentioned in post #9?