User:Narendra Sisodiya/mechanics

Deriving the equations in vectors

• All constant are taken in capital letters
• All variable are given in small letter

Deriving v(t) = Ui + A (t - Ti)

${\displaystyle {\vec {a}}={\frac {d{\vec {v}}}{dt}}={\frac {d^{2}\ {\vec {s}}}{dt^{2}}}}$

so <br\>

${\displaystyle {\vec {v}}(t)=\int _{T_{i}}^{t}{\vec {a}}dt}$

for zero or constant acceleration A we have

${\displaystyle {\vec {v}}(t)={\vec {A}}\int _{T_{i}}^{t}dt}$
${\displaystyle {\vec {v}}(t)={\vec {A}}\,[t]_{T_{i}}^{t}}$
${\displaystyle {\vec {v}}(t)={\vec {A}}\,(t-{T_{i}})+K}$

we have one unknown K , we need to consider initial or final condition

• Lets take initially we have ${\displaystyle {\vec {v}}(T_{i})={\vec {U}}_{i}}$
${\displaystyle {\vec {U}}_{i}={\vec {A}}\,({T_{i}}-{T_{i}})+K}$

ie

${\displaystyle k={\vec {U}}_{i}}$
 ${\displaystyle {\vec {v}}(t)={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})}$ eq ... (1)

If we take final condition in consideration then ie : ${\displaystyle {\vec {v}}(T_{f})={\vec {V}}_{f}}$

 ${\displaystyle {\vec {v}}(t)={\vec {V}}_{f}+{\vec {A}}\,({T_{f}}-t)}$ eq ... (2)

constant acceleration can be found using initial and final condition

 ${\displaystyle {\vec {A}}={\frac {{\vec {V}}_{f}-{\vec {U}}_{i}}{T_{f}-T_{i}}}}$ eq ... (3)

Deriving s(t) = Si + Ui(t-Ti) + (1/2)A (t - Ti)2

now we have

${\displaystyle {\vec {v}}(t)={\frac {d{\vec {s}}}{dt}}={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})}$
${\displaystyle {\vec {s}}(t)=\int _{T_{i}}^{t}{\Big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\Big )}{dt}}$
${\displaystyle {\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})\int _{T_{i}}^{t}{dt}+{\vec {A}}\int _{T_{i}}^{t}t\ {dt}}$
${\displaystyle {\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})\ {\Big [}t{\Big ]}_{T_{i}}^{t}+{\vec {A}}{\Bigg [}{\frac {t^{2}}{2}}{\Bigg ]}_{T_{i}}^{t}+K}$

${\displaystyle {\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})({t}-{T_{i}})+{\frac {\vec {A}}{2}}(t^{2}-{T_{i}}^{2})+K}$

for eliminating K we need either final or initial condition ie ${\displaystyle {\vec {s}}(T_{i})={\vec {S}}_{i}}$

${\displaystyle {\vec {S_{i}}}=({\vec {U}}_{i}-{\vec {A}}\,T_{i})({T_{i}}-{T_{i}})+{\frac {\vec {A}}{2}}({T_{i}}^{2}-{T_{i}}^{2})+K}$
${\displaystyle K={\vec {S_{i}}}}$
${\displaystyle {\vec {s}}(t)={\vec {S_{i}}}+({\vec {U}}_{i}-{\vec {A}}\,T_{i})({t}-{T_{i}})+{\frac {\vec {A}}{2}}(t^{2}-{T_{i}}^{2})}$
${\displaystyle {\vec {s}}(t)={\vec {S_{i}}}+{\vec {U}}_{i}({t}-{T_{i}})-{\vec {A}}({t}T_{i}-{T_{i}}^{2})+{\frac {\vec {A}}{2}}(t^{2}-{T_{i}}^{2})}$
${\displaystyle {\vec {s}}(t)={\vec {S_{i}}}+{\vec {U}}_{i}({t}-{T_{i}})+{\frac {\vec {A}}{2}}(t^{2}-2{t}T_{i}+{T_{i}}^{2})}$
 ${\displaystyle {\vec {s}}(t)={\vec {S_{i}}}+{\vec {U}}_{i}({t}-{T_{i}})+{\frac {\vec {A}}{2}}(t-T_{i})^{2}}$ eq ... (4)

If we would have taken final condition ie ${\displaystyle {\vec {s}}(T_{f})={\vec {S}}_{f}}$ then

 ${\displaystyle {\vec {s}}(t)={\vec {S_{f}}}+{\vec {V}}_{f}({T_{f}}-{t})-{\frac {\vec {A}}{2}}(T_{f}-t)^{2}}$ eq ... (5)

Taking ${\displaystyle t=T_{f}}$ and putting eq (3) in eq (4) , we will get

${\displaystyle {\vec {s}}(T_{f})={\vec {S_{i}}}+{\vec {U}}_{i}({T_{f}}-{T_{i}})+{\frac {\frac {{\vec {V}}_{f}-{\vec {U}}_{i}}{T_{f}-T_{i}}}{2}}(T_{f}-T_{i})^{2}}$

Or

${\displaystyle {\vec {S_{f}}}={\vec {S_{i}}}+{\vec {U}}_{i}({T_{f}}-{T_{i}})+{\frac {{\vec {V}}_{f}-{\vec {U}}_{i}}{2}}(T_{f}-T_{i})}$
 ${\displaystyle {\vec {S_{f}}}={\vec {S_{i}}}+{\frac {({\vec {V}}_{f}+{\vec {U}}_{i})(T_{f}-T_{i})}{2}}}$ eq ... (6)

Deriving |v(t)|2 = |Ui|2 + 2 A . (s(t)-S_i)

${\displaystyle {\vec {v}}(t)={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})}$
${\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\big )}.{\big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\big )}}$
${\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+{\vec {A}}.{\vec {A}}\ (t-T_{i})^{2}+2{\vec {A}}.{\vec {U}}_{i}(t-T_{i})}$
${\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+{\vec {A}}.{\vec {A}}\ (t-T_{i})^{2}+2{\vec {A}}.{\Bigg (}{\vec {s}}(t)-{\vec {S}}_{i}-{\frac {\vec {A}}{2}}(t-T_{i})^{2}{\Bigg )}}$
${\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+{\vec {A}}.{\vec {A}}\ (t-T_{i})^{2}+2{\vec {A}}.({\vec {s}}(t)-{\vec {S}}_{i})-2{\vec {A}}.({\frac {\vec {A}}{2}}(t-T_{i})^{2})}$

${\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+2{\vec {A}}.({\vec {s}}(t)-{\vec {S}}_{i})}$

taking case for final velocity

 ${\displaystyle {\vec {V}}_{f}.{\vec {V}}_{f}={\vec {U}}_{i}.{\vec {U}}_{i}+2{\vec {A}}.({\vec {S}}_{f}-{\vec {S}}_{i})}$ eq ... (7)

or

 ${\displaystyle \left|{\vec {V}}_{f}\right|^{2}=\left|{\vec {U}}_{i}\right|+2{\vec {A}}.({\vec {S}}_{f}-{\vec {S}}_{i})}$ eq ... (7)