The DMV manual says that      The faster you go, the less time you   have to avoid a hazard or collision.   The force of a 60 mph crash is not   just twice as great as a 30 mph crash;   it’s four times as great!   My physics is quite rusty, so I could not figure it out. I guess the above statement is correct, but how do we prove it?  Edit  I figured this out myself, but alternative methods or new ways of understanding are still welcome.
4034
As  says:     [...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.   Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?  My intuition is wrong here, please help it out!
103682
I'm not looking for how it's derrived. I'm wondering why it works out this way that doubling the speed quadrouples the KE.   Here's my thought process:   A car is moving down the road (ignoring air resistance and any other external force reliant on speed) at 10 m/s. To get to 20 m/s the engine should burn x amount of gas, containing x amount of energy. To do the same acceleration over again it should burn x gas again, resulting in 2x gas burned total and resulting in another 10 m/s increase in speed. This would leave it at 30 m/s.   Say the amount of gas burned to increase speed by 10 m/s is 100 energy units, and the car's mass is 2. Then increasing from 10 to 20 with ke = 1/2 mv2 should increase the KE by 100. Therefore from 20 to 30 it should also be 100, resulting in 200 total energy units and 2x gas burned. But using the equation and plugging in the same mass value with the total acceleration from 10 to 30 (30 -- 10 = 20 and plugging that in for v rather than 10) you get the total ke as 400 units rather than 200, but you theoretically should have still used 2x gas.  If someone could explain to me what the issue is in my reasoning and explain why speed is squared, that would be much appreciated.
3124
According to the Third Newton's law of motion:        For every action there is an equal and opposite reaction.   So, I understand that if I hit a brick wall with $50\, \mathrm{lbs}$ of force, the brick wall also hits me with $50\, \mathrm{lbs}$ of force (usually painfully). In this instance, Newton's third law makes sense. What I'm confused about is, if I hit a patch of drywall with $50\, \mathrm{lbs}$ of force, it's probably going to break, and due to the lack of pain in my hand, I can tell it did not hit me back with $50\, \mathrm{lbs}$ of force.   How does Newton's third law apply to situations when one object or the other is destroyed? It certainly seems like at that point it is incapable of delivering the full force of my blow back to me. What happens with the energy?
105915
In a  the OP asked why high speed will not kill us. The , highly upvoted, stated very first that      Speed doesn't kill us, but acceleration does.   The  (also well upvoted) concurs:     The danger comes from acceleration, not velocity.   Is that true?  Two comments by  and  somewhat debunk it. but they are only comments. They could not be answers, as the question ""  was not about acceleration. I wrote this question, and my answer to it (below), before I had read these comments. Like many users, I do skip a lot of comments. When I finally saw these comments already made, I wondered whether I should persist with this question. The point seems important enough to justify it, considering the high popularity of this topic.
1059
Why is speed and acceleration negative when $V_1$ of an object is say 150m/s, $V_2$ is 0 m/s and $\Delta d=0.50\,\rm m$? I found the time it takes which is 0.0033s and the acceleration to be 90909.09 m/s$^2$.   My questions:   What do negative velocity and negative acceleration mean?  Is it negative because it is slowing down? Why is the acceleration so big? Is that even reasonable?
612
The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space.
90646
I've never understood why the conservation of momentum law is taught in schools as "every force has an equal opposite reaction".   To me a gun's recoil is self explanatory; the explosion sits in-between the bullet and the back of the barrel and the gun and bullet fly in opposite directions (as would be expected). The bullet has a much higher velocity than the gun because of its lower comparative mass. Am I wrong here?  Same with stepping off a boat. You push against the boat and the mass differential between your body and the boat as well as the friction in the water enables you to jump off the boat onto the ground without falling on your face. The boat comparatively moves in the opposite direction minimally, but not because of some weird law, simply because you pushed it back!  To tell kids in school "every force has an equal and opposite force" is misleading I think. It makes it seem as if an equal mysterious force manifests out of thin air to keep the universe happy.   Just thinking - "direction of application" is only relevant to perspective, there really is no direction of application since the force and its counterpart occur simultaneously… so more accurately:   The total force can never be propagated along a single vector?
86198
I am a bit confused regarding how car moves. Following is the explanation which I've come up with after reading but I would like if someone verifies and corrects it.     If a car is moving, the rotation of it's tyre will exert a backward force on the road. Due to newton's third law, the road also exerts a force on the tyre. The force will be in the forward direction. This force causes the tire and hence the car to move forward.
13567
The Flash goes from zero to ??? in an eye blink.  As he does not have a Saturn 5 rocket on his back, how do you explain the acceleration?  How do you explain his ability to stop on a dime and turn corners without the effects of inertia?  How do you explain his alternate effect on the world around (lack of friction heat)?  How do you explain his ability to perceive so fast that he does not bump into things, can calculate where to turn flawlessly at such speed, etc?   Does he move normally in a massively slowed time-warp? Do his muscle fibers twitch at an impossibly high rate? Would his brain have to process incoming information at a speed far beyond normal consciousness, and wouldn't that make him a super-genius by normal standards? Why doesn't at least his suit heat up and burst into flame from air friction? Same for his boots and normal friction?   Or is it all "magic" and we are just supposed to accept that it "just happens?"
59488
This is quoted from A.P.French's Vibrations &amp; Waves:  If the driving force is of low frequency  relative to the natural frequency, we would expect the particle to move essentially with the driving force. This is equivalent in saying that $m\dfrac{d^2x}{dt^2}$ in $$m\dfrac{d^2x}{dt^2} + kx = F_0\cos\omega t$$ plays small role compared to the term $kx$. The amplitude is controlled by spring constant. On the other hand, at frequencies very large compared to the natural frequency, the opposite situation holds. The $kx$ becomes small compared to $m\dfrac{d^2x}{dt^2}$. In this case, we expect a relatively smaller amplitude of oscillation than the above case.  $\bullet$ Why does $kx$ become prominent in the first case &amp; not in the second case? $\bullet$ What is the physical reason behind that the first case has greater amplitude &amp; the second one smaller?
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I was waiting on a red light the other day and was wondering.    If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral?  I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed.
55539
Why is it that if I'm sitting on a seat on a bus or train and its moving quite fast, I am able to throw something in the air and easily catch it? Why is it that I haven't moved 'past' the thing during the time its travelling up and down?
88423
If 7 accidents happen in a week, what is the probability that exactly 2 accidents happen in any day of the week.  I tried to work it this way.  2 accidents out of 7 can be taken in ${7}\choose{2}$$= 21$.  Now this can happen in any of the 7 days, so we have $7 \times 21$.  Now, the remaining 5 can happen in $6^5$ ways.  So the probability is  $\frac{7\times 21 \times 6^5}{7^7} = \frac{3 \times 6^5}{7^5}$   But this gives me a value greater than 1.   I know I am doing something wrong.  Can anyone help me ?
104997
I am working on a car simulation game and I need to know how to calculate a vehicle's velocity if the gas pedal was floored and the car started accelerating from rest with some slippage in the wheels. So, is the traction force completely gone, or it decreases? And how long will the wheels keep slipping till it rotates again? Given the angular velocity of wheels, mass of car, and both the static and kinetic friction coefficients.
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Extracted from  (see the bottom section):     Some roller-coasters have loop-the-loop sections, where you travel in a vertical circle, so that you're upside-down at the top. To do this without falling off the track, you have to be traveling at least a particular minimum speed at the top. The critical factor in determining whether you make it completely around is the normal force; if the track has to exert a downward normal force at the top of the track to keep you moving in a circle, you're fine, but if the normal force drops to zero you're in trouble.   I've been told the normal force is a contrapositive force. I mean, when I'm at the top of the loop, the weight already points down so how can there be a normal force, if I'm not exerting a force to the road?
72397
I don't fully understand what would happen if we could travel at the speed of light. But I saw somewhere here that it would mean events happen out of order. But why is this a problem. It is said that cause has to happen before the effect, but why does this have to be linear?   And why is it that the speed of light is the maximum? I take it that light (photons?) are massless, so why can't they travel faster?
65843
I know that when riding close to another cyclist, I expend less energy to go the same speed. But does the Conservation of Energy "law" mean that the lead cyclist is going to expend more energy to allow me to expend less?  (i.e. Is there a physical reason to get angry at someone drafting behind you? Does riding in their slipstream cause drag for them?)