Find the initial velocity ##U##

  • #1
chwala
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Homework Statement
see attached.
Relevant Equations
Mechanics
Now in determining the initial velocity;

in my understanding, if ##s=1.8## then we consider the stone's motion from the top to the ground. Why not consider ##s=3.6##, the total distance traveled by stone from start point ##t=0##? Is it possible to model equations from this point?

The stone when thrown upwards will reach a point where it is instantaneously at rest and then start the descent. In that case, it is clear that ##s=1.8##. I need insight on this very part.

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  • #2
chwala said:
Why not consider ##s=3.6##, the total distance traveled by stone from point ##t=0##?
Because in the kinematic equations the relevant variable is the displacement (a vector) not the distance (a scalar).
 
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  • #3
Thank @kuruman I am self-studying Mechanics. Noted.
 
  • #4
chwala said:
Thank @kuruman I am self-studying Mechanics. Noted.
Then I will add here a calculation that illustrates the point in post #2.

Problem
A stone is thrown straight up in the air with initial velocity ##v_0=3~##m/s and returns to the point at which it was launched. Find the time time of flight ##T##.

Solution
The height of the stone ##y## at any time ##t## above the point of launch is given by $$y=v_0~t-\frac{1}{2}g~t^2.$$ At the specific time ##t=T## when it returns to the launch point, the height of the stone above ground is zero. With ##t=T## the equation we have $$ 0=v_0~T-\frac{1}{2}g~T^2\implies T(v_0 -\frac{1}{2}g~T)=0.$$ One or the other of the terms in the product on the right must be zero.

A. ##T=0## which says that the stone is at zero height when it is launched, a fact that we already knew and built in the equation. We reject this solution because we are looking for a time after launch, i.e. ##T>0.##

B. ##v_0 -\frac{1}{2}g~T=0 \implies T=\dfrac{2v_0}{g}.## This is the solution that we want.

Answer
##T=\dfrac{2\times 3~(\text{m/s})}{10~(\text{m/s}^2)}=0.6~\text{s}.##

You can see from this example that the total distance traveled up and down does not enter the picture and is not needed. Good luck with your self-study. We are here to help.
 
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FAQ: Find the initial velocity ##U##

What is initial velocity?

Initial velocity is the velocity of an object at the start of a time period or the velocity with which an object begins its motion. It is often denoted as ##U## or ##v_0##.

How do you calculate initial velocity in projectile motion?

In projectile motion, the initial velocity can be calculated using the equations of motion. If the angle of projection and the total time of flight are known, the initial velocity ##U## can be found using the formula: ##U = \frac{R}{T \cos(\theta)}##, where ##R## is the range, ##T## is the time of flight, and ##\theta## is the angle of projection.

What is the formula for initial velocity in uniformly accelerated motion?

For uniformly accelerated motion, the initial velocity ##U## can be found using the equation: ##U = V - at##, where ##V## is the final velocity, ##a## is the acceleration, and ##t## is the time.

How do you find initial velocity if you know the displacement, time, and acceleration?

If displacement (##s##), time (##t##), and acceleration (##a##) are known, the initial velocity ##U## can be calculated using the kinematic equation: ##s = Ut + \frac{1}{2}at^2##. Rearranging this equation gives: ##U = \frac{s - \frac{1}{2}at^2}{t}##.

Can initial velocity be negative?

Yes, initial velocity can be negative. The sign of the initial velocity depends on the chosen coordinate system and direction of motion. If the object is moving in the opposite direction to the positive axis, the initial velocity will be negative.

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