# V^2 = vi^2 + 2ad, is v velocity or speed?

flyingpig

## Homework Statement

A student throws a set of keys vertically upward to her sorority sister, who is in a window 2.00 m above. The second student catches the keys 2.30 s later.

(a) What was the velocity of the keys just before they were caught?

I did y(t) = 2 - 4.9t² + v₀t

2 = 0 - 4.9(2.3)² + v₀(2.3)

v₀ = 12.1m/s

Now I have two formulas, v² = v₀² + 2ad and v = v₀ + at, but they give me different answers.

v² = v₀² + 2ad

v² = (12.1)² + 2(-9.8)(2)

v² = 107.21

v = 10.4m/s

v = v₀ + at

v = 12.1 - 9.8(2.3)

v = -10.4m/s

I understand the square root will give us a negative as well, but the question is, HOW DO WE KNOW which to take? I hadn't use the v = v₀ + at, I wouldn't know whether to take positive or negative.

## Answers and Replies

Science Advisor
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Use:

v= v0 + at

to start. To find the distance, integrate this with respect to time:

$$d = \int vdt = \int v_0dt + \int atdt$$

What is a? (be careful about the sign). That will leave you with a quadratic equation with two possible solutions, only one of which is real.

Since the question asks for velocity, you have to give magnitude and direction of v.

AM

Gold Member
If you don't want to go the calculus route, you can keep the $$\pm$$ on the answer to $$v$$. One answer is simply going to be unphysical, you don't know right away which to use, you just have to figure out which is consistent with the other equations and the situation.

flyingpig
Does that mean the v in v² = v₀² + 2ad is actually speed and not velocity?

Gold Member
Does that mean the v in v² = v₀² + 2ad is actually speed and not velocity?

It's velocity. The problem is when you square something, you lose it's directionality (whether it's positive or negative). For example, whether or not v = 2m/s or v = -2m/s is lost in that equation since $$v^2 = 4{{m^2} \over {s^2}}$$ for either case.

Phrak
I had the same thoughts. I haven't looked at Newtonian mechanics in decades. So where does the bogus solution come from?

v² = v₀² + 2ad

(1/2)mv² - (1/2)mv₀²= adm

Delta K + Delta P = 0

The change in kinetic energy is equal to the negative change in gravitational potential energy. But, then I see that both solutions are valid. We can throw a ball upward and it hits the ceiling elastically, so its velocity is sign inverted but the energy balance is still conserved. So either solution is valid--not bogus at all, but only one solution is valid for freely falling objects.

The OP has given us a challenge: Can you determine which solution is the right one without knowledge of differential or integral calculus? This is a precalc question.

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Gold Member
Newton's Laws are time-reversal symmetric. You could run time backwards and nothing would seem out of the ordinary in terms of physics (a person dropping a ball off a cliff could be seen as someone having a ball thrown up at them). So you have to go based on what you know physically what must happen.

Phrak
OK, can you use your time reversal symmetry to recover the initial conditions...

An object is tossed vertically upward to you. Did you catch it as it was coming up at you at 10.4 m/sec, or did you catch it after it passed you and comes back down at -10.4 m/sec?

This is the dilemma of flyingpig. You don't get to use calculus. You are just given a canned set of equations.

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Gold Member
OK. An object is tossed vertically upward to you. Did you catch it as it was coming up at you at 10.4 m/sec, or did you catch it after it passed you and comes back down at -10.4 m/sec?

This is the delima of flyingpig. You don't get to use calculus. You are just given a canned set of equations.

Not exactly, although I missed the point too. What you're saying is what happens when you have 2 final velocities. This happens when you aren't given the time. In this case, we know the time, but this will result in 2 different initial velocities. Actually what I said about the answers being unphysical is not correct. Unphysical means impossible but in actuality, the two results really only mean you have 1 solution that isn't in the domain of the problem (ie. a negative time, or a "return" trip if the question were about the keys being thrown up and caught as soon as possible). So yah, not unphysical, just impossible to determine which is needed simply through the math.

Staff Emeritus
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The solution to the equation v2 = 107.21 is v = ±√(107.21).

That is, there are TWO possible answers.

This can be shown in a number of ways.

Two are:

1: v2 = 107.21

→ v2 – (√(107.21))2 = 0

→ (v – √(107.21))·(v + √(107.21)) = 0

→ v = √(107.21), or v = –√(107.21)

2: v2 = 107.21

→ √(v2) = √(107.21)

→ | v | = √(107.21)

→ v = √(107.21), or –v = √(107.21)

Yes, it's true that $$\sqrt{x^2}=\left|x\right|\,.{$$

flyingpig
The solution to the equation v2 = 107.21 is v = ±√(107.21).

That is, there are TWO possible answers.

This can be shown in a number of ways.

Two are:

1: v2 = 107.21

→ v2 – (√(107.21))2 = 0

→ (v – √(107.21))·(v + √(107.21)) = 0

→ v = √(107.21), or v = –√(107.21)

2: v2 = 107.21

→ √(v2) = √(107.21)

→ | v | = √(107.21)

→ v = √(107.21), or –v = √(107.21)

Yes, it's true that $$\sqrt{x^2}=\left|x\right|\,.{$$

But to decide which, that is my question.

flyingpig
Oh my god, I just realized something.

The v in $$v^2 = v_{0}^2 + 2a\Delta_{x}$$ is speed.

And I have a cheap proof

Let F = ma and therefore F/m = a

Then we have $$v^2 - v_{0}^2 = 2\frac{F}{m}\Deltax$$

$$\frac{1}{2}m(v^2 - v_{0}^2) = F\Delta x$$

$$\Delta KE = F\Delta x$$

clearly, we know that v in KE must be speed and this is why $$v^2 = v_{0}^2 + 2a\Delta x$$ will never give me the answer.

Am I right or what? Cmon!

Gold Member
No. The v in $${{mv^2}/2}$$ is the velocity as well. V is velocity, that's it. Stop trying to convince yourself that it is speed. Use the other kinematics you know.

$$v(t) = v_0 + at$$

The acceleration can be positive or negative. Speeds are strictly positive. If v(t) were indeed a speed, how could you ever account for what will eventually happen with all negative accelerations?

flyingpig
No. The v in $${{mv^2}/2}$$ is the velocity as well. V is velocity

But that thing $${{mv^2}/2}$$ is Kinetic energy, v is speed, it says so in my book

Science Advisor
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hi flyingpig!
I understand the square root will give us a negative as well, but the question is, HOW DO WE KNOW which to take? I hadn't use the v = v₀ + at, I wouldn't know whether to take positive or negative.

for a given vo, both solutions are correct …

if the lower student throws the keys upward with speed vo, the upper student can catch them on the way up or on the way down …

the speed v will be the same, won't it?!

if you only use v2 = vo2 + 2ad, you have no input as to t, so both solutions are correct

to eliminate one, you need to specify t, which means using v = vo + at

flyingpig
No. The v in $${{mv^2}/2}$$ is the velocity as well. V is velocity, that's it. Stop trying to convince yourself that it is speed. Use the other kinematics you know.

Sorry for bringing it up after so long.

the $$\frac{mv^2}{2}$$ is KE, it must be speed, velocity is a vector and multiplying two vectors must mean you must take either a dot or cross product, and I don't think KE involves either, so it must be speed.

A breakthrough!

flyingpig
hi flyingpig!

for a given vo, both solutions are correct …

But my book says it is wrong...

Gold Member
I'm going to pop a vain. It IS velocity, but if we call speed S, $$S = |{\bf V}|$$. $$S^2 = |{\bf V}|^2$$. However, since the velocity V is real, $$S^2 = |{\bf V}|^2 = V^2$$. Same thing. It's velocity.

Science Advisor
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hi flyingpig!
But my book says it is wrong...

yes, it's wrong

i said that for a given v0 (without specifying t), both solutions are correct …

you need to specify t also, to choose the correct solution
for a given vo, both solutions are correct …

if the lower student throws the keys upward with speed vo, the upper student can catch them on the way up or on the way down …

the speed v will be the same, won't it?!

if you only use v2 = vo2 + 2ad, you have no input as to t, so both solutions are correct

to eliminate one, you need to specify t, which means using v = vo + at

flyingpig
How can it both be correct? Going up and down isn't the same thing.

Also, just arguing back. If v was velocity then how could KE be a scalar?

Science Advisor
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How can it both be correct? Going up and down isn't the same thing.

for these purposes, they are …

she can catch them either going up or going down, and they'll be at the same speed!
Also, just arguing back. If v was velocity then how could KE be a scalar?

not following you

flyingpig
$$KE = \frac{1}{2}mv^2$$

KE is a scalar, but there is a v^2 factor, if v was velocity then how could KE be a scalar? There is no "vector product" involved

Science Advisor
Homework Helper
oh i see!

it's a dot-product …

KE = 1/2 mv.v

flyingpig
There is no dot product. I thought of that before, but I realize it was just an answer to satisfy myself. In all of the work problems with energy, it always say "speed", not velocity

Staff Emeritus
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$$\text{For vector, }\vec{v},\ \ \text{ the quantity }v^2 \text{ can be defined as: }v^2=\vec{v}\,\cdot\,\vec{v}.$$

This is of course a scalar. (Scalar Product?)

flyingpig

wikipedia said:
For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a body with constant mass m\;, whose center of mass is moving in a straight line with speed v\;, as seen above is equal to

$$E_t =\tfrac{1}{2} mv^2$$

where:

$$m\;$$ is the mass of the body
$$v\;$$ is the speed of the center of mass of the body.

What!?

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Science Advisor
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What!?

why are you ignoring the passage i linked to?

flyingpig
No I did read it and I quoted from the link that contradicts it

Staff Emeritus
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No I did read it and I quoted from the link that contradicts it
Where's the contradiction ?

flyingpig
$$v\;$$ is the speed of the center of mass of the body.

Staff Emeritus
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What does that contradict?

Stating something is a way that's different than the way it's stated somewhere else, doesn't necessarily mean that there is a contradiction.

I may re-read this whole thread again. If I think I can actually find what you're objecting to, I may attempt to clear it up for you.

flyingpig
No it clearly states that v is speed.

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"it" ?? which it?