Find the velocity of the stone just before it hits the ground

In summary, to find the velocity of a stone just before it hits the ground, apply the principles of physics, specifically using the equations of motion under uniform acceleration due to gravity. By considering the initial velocity (which is typically zero if dropped), the height from which it is dropped, and the acceleration due to gravity (approximately 9.81 m/s²), one can calculate the final velocity using the formula \( v = \sqrt{2gh} \), where \( v \) is the final velocity, \( g \) is the acceleration due to gravity, and \( h \) is the height.
  • #1
chwala
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Homework Statement
Highlighted question in Red.
Question.5
Relevant Equations
Mechanics
1710151525969.png


In my lines i have,

##h(t)=ut+\dfrac{1}{2}at^2## where ##u=0##,

with ##a=10##,

##h = \dfrac{1}{2}×10t^2## ....................1

...also,

##h-14.7 = \dfrac{1}{2}×10 (1.4-t)^2## ..................2

1- 2 gives,

##14.7 = 14t -9.8##

##24.5=14t##

##t=1.75##

Therefore, ##v = 10× 1.75=17.5## m/s , with ##u=0##.

correct? Cheers
 
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  • #2
That solution is incomprehensible to me.
 
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  • #3
If by "u" you mean initial speed, it is not zero. The stone is "thrown downwards".
 
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  • #4
What is ##h## in the equation you wrote ##~h-14.7 = \dfrac{1}{2}×10 (1.4-t)^2~?##
Hmm ##\dots## Let's see.
At ##~t=0##, ##h=(14.7+\dfrac{1}{2}\times 10\times 1.4)~\text{m}=21.7~\text{m}.##
At ##~t=1.4~\text{s}##, ##h=14.7~\text{m}.##
This doesn't match what is given.
 
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  • #5
I agree with @nasu . Initial velocity would be ##0## if the stone had been dropped or had somehow fallen.
 
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  • #6
.
 
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  • #7
Lnewqban said:
The calculated velocity can’t be correct ..
I think it is!
 
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  • #8
It's the final velocity who is 17.5 m/s and not the initial. The question asks for the final velocity (when it hits the ground). vo is about 3.5 m/s.
 
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  • #9
Oops. Sorry about that. I fixed in my mind that it was the initial velocity. I deleted the post to avoid unnecessary confusion.
 
  • #10
kuruman said:
Let's test if it is by writing a more conventional SUVAT equation and substituting the given numbers and the purported solution.

The height of the stone above ground at any time ##t## is
##h=h_0+v_0t-\frac{1}{2}gt^2##
With ##h_0=14.7~##m, ##g=10~\text{m/s}^2## and an assumed ##v_0=-17.5~##m/s, the height above ground at specific time ##t=1.4~##s is predicted to be
##h=(14.7-15.7\times1.4-\frac{1}{2}\times 10\times 1.4^2)~\text{m}=-29~\text{m}.##
That is 29 m below ground.
That's equally incomprehensible to me.

The average velocity is ##10.5m/s##, which would be at ##t = 0.7s##. So, initial velocity is ##3.5m/s## and final velocity ##17.5 m/s##.

All downwards.
 
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  • #11
PeroK said:
That's equally incomprehensible to me.

The average velocity is ##10.5m/s, which would be at ##t = 0.7s##. So, initial velocity is ##3.5m/s## and final velocity ##17.5 m/s##.
See post #9.
 
  • #12
Just to redeem myself, another way to approach this problem is to use the SUVAT equation with the final velocity replacing the initial velocity, $$h=h_0+v_f~t-\frac{1}{2}at^2.$$Of course here ##a=-g##.
 
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  • #13
nasu said:
It's the final velocity who is 17.5 m/s and not the initial. The question asks for the final velocity (when it hits the ground). vo is about 3.5 m/s.
Thank you both, @nasu and @PeroK
Deleted post with incorrectly assumed velocity.
 
  • #14
In my reasoning, in reference to my initial velocity being ##0##, the person throwing the stone had the clock ticking for some time (stone still on his hands),that is for##1.75-1.4## seconds before throwing ball ... that landed ##1.4## seconds later on the ground... that was my reasoning and it looks like it isn't correct.
 
  • #15
WWGD said:
I agree with @nasu . Initial velocity would be ##0## if the stone had been dropped or had somehow fallen.
You see 'dropped and thrown' ...to me I have to try and understand the language. I think now I see the two aren't the same.
 
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  • #16
nasu said:
It's the final velocity who is 17.5 m/s and not the initial. The question asks for the final velocity (when it hits the ground). vo is about 3.5 m/s.
Boss I was wondering how you got the ##u = 3.5##m/s.

arrrggh I see that...


From
...
##14.7 = 1.4u +9.8##
Which means,

##v^2= 3.5^2 + (2 •10•14.7)##

##v^2 = 306.15##

##v= 17.5##


Cheers man!
 
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  • #17
kuruman said:
another way to approach this problem is to use the SUVAT equation with the final velocity replacing the initial velocity,
Indeed.
The way I describe SUVAT is that there are five standard variables, s, u, v, a, t (hence the name), where u and v are initial and final velocities.
Correspondingly, there are five standard equations, each omitting one of the variables. So the first step is to identify which three you know the values of and which one you want to find, then pick the equation using those four.
It gets trickier when connected with other movements. E.g. you might know s and u only and want to find a. If the same time is involved in another process then the four relevant variables are s, u, a, t.
 

FAQ: Find the velocity of the stone just before it hits the ground

What is the initial velocity of the stone?

The initial velocity of the stone is the speed at which it is thrown or dropped. If the stone is dropped from rest, the initial velocity is 0 m/s.

What is the height from which the stone is dropped or thrown?

The height from which the stone is dropped or thrown is crucial for calculating the final velocity. This height is typically measured from the point of release to the ground.

What is the acceleration due to gravity?

The acceleration due to gravity (g) is approximately 9.8 m/s² on the surface of the Earth. This value is used in the equations of motion to determine the final velocity of the stone.

Which equation of motion should be used to find the final velocity?

The appropriate equation of motion to find the final velocity (v) of the stone just before it hits the ground is: v² = u² + 2gh, where u is the initial velocity, g is the acceleration due to gravity, and h is the height.

How do you calculate the final velocity of the stone?

To calculate the final velocity of the stone just before it hits the ground, you can use the equation: v = √(u² + 2gh). Substitute the known values for initial velocity (u), acceleration due to gravity (g), and height (h) to find the final velocity (v).

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