Put in English language with simple solution and clear
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Put in English language with simple solution and clear
ScanSolve AI Answer
Answer
2594\hat{i} + 96\hat{j} + 82944\hat{k} m/s
Here's the solution to your first question:
Step 1: Simplify the position vector.
First, combine the terms in the position vector:
r(t)=(6t3+2t)i^+(t2+3t2)j^+12t4k^r(t)=(6t3+2t)i^+(4t2)j^+12t4k^
Step 2: Find the velocity vector (v(t)).
Velocity is the first derivative of the position vector with respect to time (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 3: Find the acceleration vector (a(t)).
Acceleration is the first derivative of the velocity vector with respect to time (t):
a(t)=dtdva(t)=dtd(18t2+2)i^+dtd(8t)j^+dtd(48t3)k^a(t)=(36t)i^+(8)j^+(144t2)k^
Step 4: Calculate velocity and acceleration at t=12 sec.
Substitute t=12 into the velocity vector:
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^
Substitute t=12 into the acceleration vector:
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 5: Calculate velocity at t=10 sec.
Substitute t=10 into the velocity vector:
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 6: Find the difference in velocity between t=12 and t=10.
Difference = v(12)−v(10)Difference=(2594i^+96j^+82944k^)−(1802i^+80j^+48000k^)Difference=(2594−1802)i^+(96−80)j^+(82944−48000)k^Difference=792i+16j^+34944k^m/s^
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This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.
ScanSolve AI Answer
Here's the solution to your first question: Step 1: Simplify the position vector. First, combine the terms in the position vector: r(t) = (6t^3 + 2t)i + (t^2 + 3t^2)j + 12t^4k r(t) = (6t^3 + 2t)i + (4t^2)j + 12t^4k Step 2: Find the velocity vector (v(t)). Velocity is the first derivative of the position vector with respect to time (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 3: Find the acceleration vector (a(t)). Acceleration is the first derivative of the velocity vector with respect to time (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 4: Calculate velocity and acceleration at t = 12 sec. Substitute t = 12 into the velocity vector: 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 Substitute t = 12 into the acceleration vector: 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 5: Calculate velocity at t = 10 sec. Substitute t = 10 into the velocity vector: 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 6: Find the difference in velocity between t = 12 and t = 10. Difference = v(12) - v(10) Difference = (2594i + 96j + 82944k) - (1802i + 80j + 48000k) Difference = (2594 - 1802)i + (96 - 80)j + (82944 - 48000)k Difference = 792i + 16j + 34944k m/s That's 2 down. 3 left today — send the next one.
