# The Naked Scientists Forum

### Author Topic: How do I calculate the acceleration of the bullet?  (Read 19244 times)

#### Vicky Garg

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##### How do I calculate the acceleration of the bullet?
« on: 06/02/2011 23:30:03 »
Vicky Garg asked the Naked Scientists:
A bullet moving with velocity of 10m/s is brought to rest after penetrating the wooden plank of 4cm thickness.  How do I calculate the acceleration of the bullet?
What do you think?
« Last Edit: 06/02/2011 23:30:03 by _system »

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #1 on: 07/02/2011 01:46:24 »
You might try Googling "Equations of Motion".

#### williampcochran

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##### How do I calculate the acceleration of the bullet?
« Reply #2 on: 07/02/2011 21:08:39 »
vf = final velocity
vi = initial velocity
a  = acceleration
t  = time

#### williampcochran

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##### How do I calculate the acceleration of the bullet?
« Reply #3 on: 07/02/2011 21:10:28 »
is this a high school physics question?

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #4 on: 08/02/2011 07:11:51 »
is this a high school physics question?

What!  [:0]  [:0]  [:0]  [:0]

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #5 on: 08/02/2011 13:24:01 »
Vicky Garg asked the Naked Scientists:
A bullet moving with velocity of 10m/s is brought to rest after penetrating the wooden plank of 4cm thickness.  How do I calculate the acceleration of the bullet?
What do you think?
You can't, unless you already know, or you make hypothesis, about how the force (and so the acceleration) depends on the space traveled inside the plank (or how it depends on time).

If you make, for example, the simplest hypothesis that the force is constant, so the acceleration is constant = a, then s = 1/2 v2/a where s = space traveled by the bullet, v = bullet initial speed. Then a = v2/2s = 100/0.08 = 1250 m*s-2 = 127.55g.

With different informations about the force, the problem is more complicated, assuming that, however, you are interested in the *average* acceleration.
« Last Edit: 08/02/2011 13:28:03 by lightarrow »

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #6 on: 08/02/2011 22:42:46 »
Vicky Garg asked the Naked Scientists:
A bullet moving with velocity of 10m/s is brought to rest after penetrating the wooden plank of 4cm thickness.  How do I calculate the acceleration of the bullet?
What do you think?
You can't, unless you already know, or you make hypothesis, about how the force (and so the acceleration) depends on the space traveled inside the plank (or how it depends on time).

If you make, for example, the simplest hypothesis that the force is constant, so the acceleration is constant = a, then s = 1/2 v2/a where s = space traveled by the bullet, v = bullet initial speed. Then a = v2/2s = 100/0.08 = 1250 m*s-2 = 127.55g.

With different informations about the force, the problem is more complicated, assuming that, however, you are interested in the *average* acceleration.

Based on the data, and applying v˛ = u˛+2as, we find that,

a = -u˛/2s

or -1250 m/s˛

That's the acceleration, regardless of the the force. We only need to know the initial velocity (u) the final velocity (v) and the distance travelled (s) to determine the acceleration (a).  The possibility that the acceleration was not uniform only means that we can't quantify the maximum or minimum acceleration.

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #7 on: 09/02/2011 16:22:47 »
Based on the data, and applying v˛ = u˛+2as,
And how do you get this without making the assumption that the motion is uniformly accelerated?

#### imatfaal

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##### How do I calculate the acceleration of the bullet?
« Reply #8 on: 09/02/2011 16:56:58 »
you make no assumption that acceleration is uniform - just that over a distance the acceleration will be...  you choose your equation of motion and you take your choice
you got v, you got u, and you got s, and you want a
v^2=u^2+2as
seems a pretty good choice

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #9 on: 09/02/2011 18:01:41 »
you make no assumption that acceleration is uniform - just that over a distance the acceleration will be...  you choose your equation of motion and you take your choice
you got v, you got u, and you got s, and you want a
v^2=u^2+2as
seems a pretty good choice
No.

Let's make an example.
Someone says he has good reasons to assume that the force can be considered proportional to the distance traveled x, because at high speeds (as in this case) wood behaves as a spring (it's just for sake of discussion, I personally have no reasons to make such an assumption, because I don't know what really happens). So F = -kx.

Then: mx'' + kx = 0

--> x(t) = A*sin(ωt) where ω = sqrt(k/m)

since x(0) = 0; x'(0) = v, you find A = v/ω. Knowing that at t = t1 the bullet stops and x(t1) = 0.04m, you find: t1 = π/2ω and: ω = v/s = 10/0.04 = 250 rad/s.

Now let's compute the average acceleration:
<x''(t)> = (v - 0)/(t1 - 0) = v/(π/2ω) = 2ω*v/π = 5000/π ≈ 1591.55 m*s-2 which is different from 1250 m*s-2.
« Last Edit: 09/02/2011 18:05:36 by lightarrow »

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #10 on: 09/02/2011 18:35:04 »
I think we did calculate the average acceleration. Average acceleration is Δv/Δt. I believe that's what we get from
v˛ = u˛+2as

Are you perhaps assuming constant acceleration?
« Last Edit: 09/02/2011 19:06:49 by Geezer »

#### imatfaal

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##### How do I calculate the acceleration of the bullet?
« Reply #11 on: 09/02/2011 18:51:44 »
Responding to Lightarrow - cos geezer responded in the meantime

So basically you are making an assumption that is not in the question.

Perhaps for extra credit you might introduce your assumptions but please note
1.  not high speed - i have met people who can run faster
2.  any reason, other than we know the formula, to model on a spring
3.  the equations governing forces on a spring are limited to being within the linear xpansion of the spring - outside those limits the standard hooke's law does not apply thus the use of a shm to model is doubly suspect

#### yor_on

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##### How do I calculate the acceleration of the bullet?
« Reply #12 on: 09/02/2011 19:18:58 »
Just a small question, when that bullet stops moving due to its initial force, 'slowing', gravitation taking over as the main 'force'. Is it then 'decelerating' or 'accelerating'? :)

#### Bored chemist

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##### How do I calculate the acceleration of the bullet?
« Reply #13 on: 09/02/2011 19:32:47 »
One aspect of the bullet and its motion specified in the question is that it is brought to rest. After that it has an acceleration (and velocity) of zero (measured WRT the plank).
I bet you wouldn't get full marks for that answer.

Would anyone like to solve the problem under the assumption that the plank is suspended from a string and is set swinging by the bullet?
Is the plank assumed to be at rest in the first place?
« Last Edit: 09/02/2011 19:35:46 by Bored chemist »

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #14 on: 10/02/2011 13:31:57 »
I think we did calculate the average acceleration. Average acceleration is Δv/Δt. I believe that's what we get from
v˛ = u˛+2as

Are you perhaps assuming constant acceleration?
And how do you compute Δt without solving the equation of motion, which you don't have at all?

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #15 on: 10/02/2011 14:42:07 »
Responding to Lightarrow - cos geezer responded in the meantime

So basically you are making an assumption that is not in the question.

Perhaps for extra credit you might introduce your assumptions but please note
1.  not high speed - i have met people who can run faster
2.  any reason, other than we know the formula, to model on a spring
3.  the equations governing forces on a spring are limited to being within the linear xpansion of the spring - outside those limits the standard hooke's law does not apply thus the use of a shm to model is doubly suspect
No, it's you Geezer that makes the arbitrary assumption that the motion is uniformly accelerated. If you don't make any assumption about the equation of motion/how the force depends on time or distance, you *cannot* solve the problem.
« Last Edit: 10/02/2011 14:49:10 by lightarrow »

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #16 on: 10/02/2011 14:46:53 »
One aspect of the bullet and its motion specified in the question is that it is brought to rest. After that it has an acceleration (and velocity) of zero (measured WRT the plank).
I bet you wouldn't get full marks for that answer.

Would anyone like to solve the problem under the assumption that the plank is suspended from a string and is set swinging by the bullet?
Is the plank assumed to be at rest in the first place?
If you allow me to make 4 further assumptions, that is:
1. the "ballistic hypothesis"(*),
2. neglect the air's resistance,
3. the plank is homogeneous,
4. the bullet impacts in the plank's centre of mass,
then I can solve it (faster if the plank is initially at rest  ). Without some of those assumptions, the problem can become quite or extremely, more difficult.

(*) means that we neglect the plank's displacement during the bullet's penetration into it; such hypothesis is valid in the limit m/M << 1
m = bullet's mass, M = plank's mass.
« Last Edit: 10/02/2011 15:00:10 by lightarrow »

#### syhprum

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##### How do I calculate the acceleration of the bullet?
« Reply #17 on: 10/02/2011 16:38:26 »
I calculated the acceleration by a rather indirect but conceptually simple manner, I visualised the bullet acquiring its velocity of 10m/s by falling in the Earths gravity and calculated how far it would have to travel.

D=V^2/2g =5.0916497 m

it is then stopped in a distance of .04m hence the average acceleration is D*g/.04=1250 m/s^2
= 127.55g

« Last Edit: 10/02/2011 16:47:05 by syhprum »

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #18 on: 10/02/2011 17:49:34 »
I think we did calculate the average acceleration. Average acceleration is Δv/Δt. I believe that's what we get from
v˛ = u˛+2as

Are you perhaps assuming constant acceleration?
And how do you compute Δt without solving the equation of motion, which you don't have at all?

Oh! That's easy.

Δt = 2s/(u+v)
= 2s/u
= 8 ms

#### Bored chemist

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##### How do I calculate the acceleration of the bullet?
« Reply #19 on: 10/02/2011 20:57:30 »
One aspect of the bullet and its motion specified in the question is that it is brought to rest. After that it has an acceleration (and velocity) of zero (measured WRT the plank).
I bet you wouldn't get full marks for that answer.

Would anyone like to solve the problem under the assumption that the plank is suspended from a string and is set swinging by the bullet?
Is the plank assumed to be at rest in the first place?
If you allow me to make 4 further assumptions, that is:
1. the "ballistic hypothesis"(*),
2. neglect the air's resistance,
3. the plank is homogeneous,
4. the bullet impacts in the plank's centre of mass,
then I can solve it (faster if the plank is initially at rest  ). Without some of those assumptions, the problem can become quite or extremely, more difficult.

(*) means that we neglect the plank's displacement during the bullet's penetration into it; such hypothesis is valid in the limit m/M << 1
m = bullet's mass, M = plank's mass.
Since I stated that the bullet sets the plank swinging you cannot assume that the plank's mass is >> that of the bullet.
Or, to put it another way, I was talking about the extremely difficult version of the problem

Seriously, I was just pointing out that there are lots of unstated assumptions.

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #20 on: 10/02/2011 22:16:28 »
Responding to Lightarrow - cos geezer responded in the meantime

So basically you are making an assumption that is not in the question.

Perhaps for extra credit you might introduce your assumptions but please note
1.  not high speed - i have met people who can run faster
2.  any reason, other than we know the formula, to model on a spring
3.  the equations governing forces on a spring are limited to being within the linear xpansion of the spring - outside those limits the standard hooke's law does not apply thus the use of a shm to model is doubly suspect
No, it's you Geezer that makes the arbitrary assumption that the motion is uniformly accelerated. If you don't make any assumption about the equation of motion/how the force depends on time or distance, you *cannot* solve the problem.

That is true. It does depend on whether or not the wood exerts a constant force on the bullet. It's probably not a completely unreasonable assumption that it does, but it would take a rather fancy setup to find out. I suppose you could mount the wood on a pressure transducer. Anybody know of a better way?

#### Bored chemist

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##### How do I calculate the acceleration of the bullet?
« Reply #21 on: 11/02/2011 07:16:50 »
It might be simple to assume that the force is constant, but it's unlikely. As the bullet slows down the force on it is likely to decrease. After all, the force is zero when the bullet's speed is zero.
You could use a pressure transducer, but a force transducer would seem better.

#### Geezer

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##### How do I calculate the acceleration of the bullet?
« Reply #22 on: 11/02/2011 07:48:30 »
It might be simple to assume that the force is constant, but it's unlikely.

It is quite likely if it's largely a function of friction between the bullet and the wood.

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #23 on: 11/02/2011 08:40:05 »

Since I stated that the bullet sets the plank swinging you cannot assume that the plank's mass is >> that of the bullet.
You are half right  : the rate m/M can be << 1 and at the same time high enough to set the plank in motion. This is what we do with the ballistic pendulum:
http://en.wikipedia.org/wiki/Ballistic_pendulum

What counts is the bullet momentum. The initial speed V of the pendulum, as a function of the bullet's speed v, is then simply V = m*v/M (momentum conservation).
Since the bullet's speed is usually high, its mass can be much smaller than that of the pendulum and V is however not negligible.

#### lightarrow

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##### How do I calculate the acceleration of the bullet?
« Reply #24 on: 11/02/2011 09:37:01 »
Concerning the force on the bullet, I wouldn't take anything for granted. Being wood, I'd expect, for example, that for a very short path the force to be high while the fibers in the cylinder in front of the bullet are being stretched and broken, then that the force decreases while the wooden cylinder is moved ahead and compressed but with little force, then that the cylinder is being compressed at high pressure and the force increases with the path. In case, instead, the bullet goes through all the plank escaping out, I expect the force increases for a short path, then decreases when the cylinder just moves inside the hole.
Anyway, I wouldn't bet on anything particularly simple.

#### The Naked Scientists Forum

##### How do I calculate the acceleration of the bullet?
« Reply #24 on: 11/02/2011 09:37:01 »