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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d91d2980bbdb75a94c762f322576a0b62cd19cee)
so <br\>
![{\displaystyle {\vec {v}}(t)=\int _{T_{i}}^{t}{\vec {a}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10770c59af5a55a725f2eaae7c76e3273c7629a4)
for zero or constant acceleration A we have
![{\displaystyle {\vec {v}}(t)={\vec {A}}\int _{T_{i}}^{t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd5f17c7bb05adce6d4f508f4732f85cb3cf03a)
![{\displaystyle {\vec {v}}(t)={\vec {A}}\,[t]_{T_{i}}^{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a9033eab74c11422ea8c844bc660a1d1a70e9b)
![{\displaystyle {\vec {v}}(t)={\vec {A}}\,(t-{T_{i}})+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b63b7fb7e302b69afb6962f4b59e8afdc2e09e5)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5277b19eaf101ae5ca9ee470de64a1ba733caf)
![{\displaystyle {\vec {U}}_{i}={\vec {A}}\,({T_{i}}-{T_{i}})+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a175aae86208cd35acdde488ff2ff084806733b7)
ie
![{\displaystyle k={\vec {U}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf6c130a6fc43b9932f92ede5df2d3cd8e331d4)
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eq ... (1)
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If we take final condition in consideration then ie :
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eq ... (2)
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constant acceleration can be found using initial and final condition
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eq ... (3)
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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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38626333e82caae643efb28ff4be27a718951228)
![{\displaystyle {\vec {s}}(t)=\int _{T_{i}}^{t}{\Big (}{\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}}){\Big )}{dt}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6707dbcab46a854b9f7c842ee9435dcecf9784b9)
![{\displaystyle {\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})\int _{T_{i}}^{t}{dt}+{\vec {A}}\int _{T_{i}}^{t}t\ {dt}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1438beb129ef3e3843e0d244b658b6c824a3350e)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e2a126438bb9d2d8f4661f75a232df50a58d15)
![{\displaystyle {\vec {s}}(t)=({\vec {U}}_{i}-{\vec {A}}\,T_{i})({t}-{T_{i}})+{\frac {\vec {A}}{2}}(t^{2}-{T_{i}}^{2})+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8778c4f2c32359537274b0ac0211891c753a7def)
for eliminating K we need either final or initial condition ie
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc31e4fb32814b8a563f2c81517de92204d85322)
![{\displaystyle K={\vec {S_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00b91855b261dbd344ca0cf6eaceee16774b336d)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a0c072e245fda227e03bf8e88ff35c6ee70392)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/653d7897a4bf32eb2b15e0aa8c6dc3b8c158bd44)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c6d63aad61bd83fae1d184071bddb820ffa200)
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eq ... (4)
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If we would have taken final condition ie
then
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eq ... (5)
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Taking
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36276b526b5e84bac4c2e271f3acc8e355cf5931)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf55a086667b9a5be284215638fed1fa74b0054)
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eq ... (6)
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Deriving |v(t)|2 = |Ui|2 + 2 A . (s(t)-S_i)
![{\displaystyle {\vec {v}}(t)={\vec {U}}_{i}+{\vec {A}}\,(t-{T_{i}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534d7ac72eb075f0dbf5e9628fd341b1eb366136)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c636e47dac89f0b4fea88a64570b83f88e6837)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6007efb930f7403df0655408a33517c6b3ea15)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e373564eabf9cb1abb2d9fe6abc72e7d0006a8)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d97583a077ee03fee9875f42c3d2b6fff641ed67)
![{\displaystyle {\vec {v}}(t).{\vec {v}}(t)={\vec {U}}_{i}.{\vec {U}}_{i}+2{\vec {A}}.({\vec {s}}(t)-{\vec {S}}_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20a52127611b94851fb4b58addbd803ee5a0bf88)
taking case for final velocity
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eq ... (7)
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or
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eq ... (7)
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