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) = U_i + A (t - T_i)

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

so <br\>

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

for zero or constant acceleration A we have

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

${\vec {v}}(t)={\vec {A}}\,[t]_{T_{i}}^{t}$

${\vec {v}}(t)={\vec {A}}\,(t-{T_{i}})+K$

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

• Lets take initially we have ${\vec {v}}(T_{i})={\vec {U}}_{i}$

${\vec {U}}_{i}={\vec {A}}\,({T_{i}}-{T_{i}})+K$

ie

$k={\vec {U}}_{i}$

${\vec {v}}(t)={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})$

eq ... (1)

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

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

eq ... (2)

constant acceleration can be found using initial and final condition

${\vec {A}}={\frac {{\vec {V}}_{f}-{\vec {U}}_{i}}{T_{f}-T_{i}}}$

eq ... (3)

Deriving s(t) = S_i + U_i(t-T_i) + (1/2)A (t - T_i)²

now we have

${\vec {v}}(t)={\frac {d{\vec {s}}}{dt}}={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})$

${\vec {s}}(t)=\int _{T_{i}}^{t}{\Big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\Big )}{dt}$

${\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})\int _{T_{i}}^{t}{dt}+{\vec {A}}\int _{T_{i}}^{t}t\ {dt}$

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

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

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

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

$K={\vec {S_{i}}}$

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

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

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

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

eq ... (4)

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

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

eq ... (5)

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

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

Or

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

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

eq ... (6)

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

${\vec {v}}(t)={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})$

${\vec {v}}(t).{\vec {v}}(t)={\big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\big )}.{\big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\big )}$

${\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+{\vec {A}}.{\vec {A}}\ (t-T_{i})^{2}+2{\vec {A}}.{\vec {U}}_{i}(t-T_{i})$

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

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

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

taking case for final velocity

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

eq ... (7)

or

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

eq ... (7)