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

## 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 = (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.

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Andrew Mason
Homework Helper
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

Pengwuino
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.

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

Pengwuino
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.

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

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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Pengwuino
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.

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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Pengwuino
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.

SammyS
Staff Emeritus
Homework Helper
Gold Member
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 $$\sqrt{x^2}=\left|x\right|\,.{$$

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.

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!

Pengwuino
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?

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

tiny-tim
Homework Helper
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 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!

hi flyingpig! for a given vo, both solutions are correct …
But my book says it is wrong...

Pengwuino
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.

tiny-tim
Homework Helper
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 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?

tiny-tim
Homework Helper
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 $$KE = \frac{1}{2}mv^2$$

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

tiny-tim
oh i see! KE = 1/2 mv.v 