The particle has constant acceleration according to
![\vec a(t)=2\,\vec\imath-4\,\vec\jmath-2\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20a%28t%29%3D2%5C%2C%5Cvec%5Cimath-4%5C%2C%5Cvec%5Cjmath-2%5C%2C%5Cvec%20k)
Its velocity at time
is
![\displaystyle\vec v(t)=\vec v(0)+\int_0^t\vec a(u)\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cvec%20v%28t%29%3D%5Cvec%20v%280%29%2B%5Cint_0%5Et%5Cvec%20a%28u%29%5C%2C%5Cmathrm%20du)
![\vec v(t)=\vec v(0)+(2\,\vec\imath-4\,\vec\jmath-2\,\vec k)t](https://tex.z-dn.net/?f=%5Cvec%20v%28t%29%3D%5Cvec%20v%280%29%2B%282%5C%2C%5Cvec%5Cimath-4%5C%2C%5Cvec%5Cjmath-2%5C%2C%5Cvec%20k%29t)
![\vec v(t)=(v_{0x}+2t)\,\vec\imath+(v_{0y}-4t)\,\vec\jmath+(v_{0z}-2t)\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20v%28t%29%3D%28v_%7B0x%7D%2B2t%29%5C%2C%5Cvec%5Cimath%2B%28v_%7B0y%7D-4t%29%5C%2C%5Cvec%5Cjmath%2B%28v_%7B0z%7D-2t%29%5C%2C%5Cvec%20k)
Then the particle has position at time
according to
![\displaystyle\vec r(t)=\vec r(0)+\int_0^t\vec v(u)\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cvec%20r%28t%29%3D%5Cvec%20r%280%29%2B%5Cint_0%5Et%5Cvec%20v%28u%29%5C%2C%5Cmathrm%20du)
![\vec r(t)=(3+v_{0x}t+t^2)\,\vec\imath+(6+v_{0y}t-2t^2)\,\vec\jmath+(9+v_{0z}t-t^2)\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20r%28t%29%3D%283%2Bv_%7B0x%7Dt%2Bt%5E2%29%5C%2C%5Cvec%5Cimath%2B%286%2Bv_%7B0y%7Dt-2t%5E2%29%5C%2C%5Cvec%5Cjmath%2B%289%2Bv_%7B0z%7Dt-t%5E2%29%5C%2C%5Cvec%20k)
At at the point (3, 6, 9), i.e. when
, it has speed 8, so that
![\|\vec v(0)\|=8\iff{v_{0x}}^2+{v_{0y}}^2+{v_{0z}}^2=64](https://tex.z-dn.net/?f=%5C%7C%5Cvec%20v%280%29%5C%7C%3D8%5Ciff%7Bv_%7B0x%7D%7D%5E2%2B%7Bv_%7B0y%7D%7D%5E2%2B%7Bv_%7B0z%7D%7D%5E2%3D64)
We know that at some time
, the particle is at the point (5, 2, 7), which tells us
![\begin{cases}3+v_{0x}T+T^2=5\\6+v_{0y}T-2T^2=2\\9+v_{0z}T-T^2=7\end{cases}\implies\begin{cases}v_{0x}=\dfrac{2-T^2}T\\\\v_{0y}=\dfrac{2T^2-4}T\\\\v_{0z}=\dfrac{T^2-2}T\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D3%2Bv_%7B0x%7DT%2BT%5E2%3D5%5C%5C6%2Bv_%7B0y%7DT-2T%5E2%3D2%5C%5C9%2Bv_%7B0z%7DT-T%5E2%3D7%5Cend%7Bcases%7D%5Cimplies%5Cbegin%7Bcases%7Dv_%7B0x%7D%3D%5Cdfrac%7B2-T%5E2%7DT%5C%5C%5C%5Cv_%7B0y%7D%3D%5Cdfrac%7B2T%5E2-4%7DT%5C%5C%5C%5Cv_%7B0z%7D%3D%5Cdfrac%7BT%5E2-2%7DT%5Cend%7Bcases%7D)
and in particular we see that
![v_{0y}=-2v_{0x}](https://tex.z-dn.net/?f=v_%7B0y%7D%3D-2v_%7B0x%7D)
and
![v_{0z}=-v_{0x}](https://tex.z-dn.net/?f=v_%7B0z%7D%3D-v_%7B0x%7D)
Then
![{v_{0x}}^2+(-2v_{0x})^2+(-v_{0x})^2=6{v_{0x}}^2=64\implies v_{0x}=\pm\dfrac{4\sqrt6}3](https://tex.z-dn.net/?f=%7Bv_%7B0x%7D%7D%5E2%2B%28-2v_%7B0x%7D%29%5E2%2B%28-v_%7B0x%7D%29%5E2%3D6%7Bv_%7B0x%7D%7D%5E2%3D64%5Cimplies%20v_%7B0x%7D%3D%5Cpm%5Cdfrac%7B4%5Csqrt6%7D3)
![\implies v_{0y}=\mp\dfrac{8\sqrt6}3](https://tex.z-dn.net/?f=%5Cimplies%20v_%7B0y%7D%3D%5Cmp%5Cdfrac%7B8%5Csqrt6%7D3)
![\implies v_{0z}=\mp\dfrac{4\sqrt6}3](https://tex.z-dn.net/?f=%5Cimplies%20v_%7B0z%7D%3D%5Cmp%5Cdfrac%7B4%5Csqrt6%7D3)
That is, there are two possible initial velocities for which the particle can travel between (3, 6, 9) and (5, 2, 7) with the given acceleration vector and given that it starts with a speed of 8. Then there are two possible solutions for its position vector; one of them is
![\vec r(t)=\left(3+\dfrac{4\sqrt6}3t+t^2\right)\,\vec\imath+\left(6-\dfrac{8\sqrt6}3t-2t^2\right)\,\vec\jmath+\left(9-\dfrac{4\sqrt6}3t-t^2\right)\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20r%28t%29%3D%5Cleft%283%2B%5Cdfrac%7B4%5Csqrt6%7D3t%2Bt%5E2%5Cright%29%5C%2C%5Cvec%5Cimath%2B%5Cleft%286-%5Cdfrac%7B8%5Csqrt6%7D3t-2t%5E2%5Cright%29%5C%2C%5Cvec%5Cjmath%2B%5Cleft%289-%5Cdfrac%7B4%5Csqrt6%7D3t-t%5E2%5Cright%29%5C%2C%5Cvec%20k)
The sharp nail has a less surface area in comparison to a blunt nail and pressure is inversely proportional to area so it is easier to Hamer a sharp nail into a wood rather than having a blunt nail in wood
Answer:
The Reynolds number determines what type of regime is laminar or turbulent in a flow pattern.
Explanation:
The Reynolds number determines what type of regime is laminar or turbulent in a flow pattern.
In the laminar regime R_D <2300 the flow is ordered and has an envelope predicted by the expressions
In the turbulent regime R_D> 2900 the flow has vortices and is extremely irregular
in the intermediate values it is in a transition regime