Find the velocity and acceleration of r=(6t3+2t)i + (t2+3t3)j + 12t4 at t=12 sec. Then find the difference of velocity at t=10 and t=12. 2. Find (P^Q) r use truth table.
|Mathematics
Find the velocity and acceleration of r=(6t3+2t)i + (t2+3t3)j + 12t4 at t=12 sec. Then find the difference of velocity at t=10 and t=12. 2. Find (P^Q) r use truth table.
ScanSolve AI Answer
Answer
2594\hat{i} + 96\hat{j} + 82944\hat{k} m/s
Hapa kuna suluhisho la swali lako la kwanza:
Step 1: Tafuta vekta ya kasi (v(t)).
Kwanza, rahisiisha vekta ya nafasi iliyopewa:
r(t)=(6t3+2t)i^+(t2+3t2)j^+12t4k^r(t)=(6t3+2t)i^+(4t2)j^+12t4k^
Kasi ni derivative ya kwanza ya nafasi kwa heshima ya muda (t):
v(t)=dtdrv(t)=dtd(6t3+2t)i^+dtd(4t2)j^+dtd(12t4)k^v(t)=(18t2+2)i^+(8t)j^+(48t3)k^
Step 2: Tafuta vekta ya kuongeza kasi (a(t)).
Kuongeza kasi ni derivative ya kwanza ya kasi kwa heshima ya muda (t):
a(t)=dtdva(t)=dtd(18t2+2)i^+dtd(8t)j^+dtd(48t3)k^a(t)=(36t)i^+(8)j^+(144t2)k^
Step 3: Kokotoa kasi na kuongeza kasi kwa t=12 sec.
Badilisha t=12 katika v(t):
v(12)=(18(12)2+2)i^+(8(12))j^+(48(12)3)k^v(12)=(18(144)+2)i^+(96)j^+(48(1728))k^v(12)=(2592+2)i^+96j^+82944k^v(12)=2594i+96j^+82944k^m/s^
Badilisha t=12 katika a(t):
a(12)=(36(12))i^+(8)j^+(144(12)2)k^a(12)=(432)i^+8j^+(144(144))k^a(12)=432i+8j^+20736k^m/s2^
Step 4: Kokotoa kasi kwa t=10 sec.
Badilisha t=10 katika v(t):
v(10)=(18(10)2+2)i^+(8(10))j^+(48(10)3)k^v(10)=(18(100)+2)i^+(80)j^+(48(1000))k^v(10)=(1800+2)i^+80j^+48000k^v(10)=1802i^+80j^+48000k^m/s
Step 5: Tafuta tofauti ya kasi kati ya t=10 na t=12.
Tofauti = v(12)−v(10)Tofauti=(2594i^+96j^+82944k^)−(1802i^+80j^+48000k^)Tofauti=(2594−1802)i^+(96−80)j^+(82944−48000)k^Tofauti=792i+16j^+34944k^m/s^
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Hapa kuna suluhisho la swali lako la kwanza: Step 1: Tafuta vekta ya kasi (v(t)).
Find the velocity and acceleration of r=(6t3+2t)i + (t2+3t3)j + 12t4 at t=12 sec. Then find the difference of velocity at t=10 and t=12. 2. Find (P^Q) r use truth table.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hapa kuna suluhisho la swali lako la kwanza: Step 1: Tafuta vekta ya kasi (v(t)). Kwanza, rahisiisha vekta ya nafasi iliyopewa: r(t) = (6t^3 + 2t)i + (t^2 + 3t^2)j + 12t^4k r(t) = (6t^3 + 2t)i + (4t^2)j + 12t^4k Kasi ni derivative ya kwanza ya nafasi kwa heshima ya muda (t): v(t) = drdt v(t) = (d)/(dt)(6t^3 + 2t)i + (d)/(dt)(4t^2)j + (d)/(dt)(12t^4)k v(t) = (18t^2 + 2)i + (8t)j + (48t^3)k Step 2: Tafuta vekta ya kuongeza kasi (a(t)). Kuongeza kasi ni derivative ya kwanza ya kasi kwa heshima ya muda (t): a(t) = dvdt a(t) = (d)/(dt)(18t^2 + 2)i + (d)/(dt)(8t)j + (d)/(dt)(48t^3)k a(t) = (36t)i + (8)j + (144t^2)k Step 3: Kokotoa kasi na kuongeza kasi kwa t = 12 sec. Badilisha t = 12 katika v(t): v(12) = (18(12)^2 + 2)i + (8(12))j + (48(12)^3)k v(12) = (18(144) + 2)i + (96)j + (48(1728))k v(12) = (2592 + 2)i + 96j + 82944k v(12) = 2594i + 96j + 82944k m/s Badilisha t = 12 katika a(t): a(12) = (36(12))i + (8)j + (144(12)^2)k a(12) = (432)i + 8j + (144(144))k a(12) = 432i + 8j + 20736k m/s^2 Step 4: Kokotoa kasi kwa t = 10 sec. Badilisha t = 10 katika v(t): v(10) = (18(10)^2 + 2)i + (8(10))j + (48(10)^3)k v(10) = (18(100) + 2)i + (80)j + (48(1000))k v(10) = (1800 + 2)i + 80j + 48000k v(10) = 1802i + 80j + 48000k m/s Step 5: Tafuta tofauti ya kasi kati ya t = 10 na t = 12. Tofauti = v(12) - v(10) Tofauti = (2594i + 96j + 82944k) - (1802i + 80j + 48000k) Tofauti = (2594 - 1802)i + (96 - 80)j + (82944 - 48000)k Tofauti = 792i + 16j + 34944k m/s Tuma swali linalofuata 📸
