An object has the 3-dimensional position functions listed below; x, y, and z are

Question

An object has the 3-dimensional position functions listed below; x, y, and z are in meters and t is in seconds.

x(t) = 6.1 + 1.8t + 4.7t2 + 0.85t3

y(t) = 8.8 + 8.2t + 1.2t2 + 0.49t3

z(t) = 8.3 + 6.4t + 3.3t2 + 0.66t3


None of the vector notation answers require units.

A. What is the initial position of the object in i, j, k notation? (   +   +  ) m

B. What is the initial velocity of the object in i, j, k notation? (   +   +  ) m/s

C. What is the initial acceleration of the object in i, j, k notation? (   +   +  ) m/s2

D. What is the position of the object at t = 4.0 s in i, j, k notation?(   +   +  ) m

E. What is the velocity of the object at t = 4.0 s in i, j, k notation? (   +   +  ) m/s

F. What is the vector  at t = 4.0 s? (   +   +  ) m2/s

Explanation / Answer

A.) Initial Position is at t= 0

x(0) = 6.1 m

y(0) = 8.8 m

z(0) = 8.3 m

Therefore initial position = (6.1i + 8.8j + 8.3 k) m

B.) Velocity is the differentiation of position with respect to time.

x' (t) = 1.8 + 9.4t + 2.55t2 => x'(0) = 1.8 m/s

y'(t)= 8.2 + 2.4 t + 1.47t2 => y'(0) = 8.2 m/s

z'(t) = 6.4 + 6.6t + 1.98t2 => z'(0) = 6.4 m/s

So, initial velocity v = (1.8i +8.2 j +6.4k) m/s

C.) Acceleration if the differentiation of velocity with respect to time.

x'' (t) = 9.4 + 5.1t => x''(0) = 9.4 m/s2  

y''(t) = 2.4 + 2.94 t => y''(0) = 2.4 m/s2

z''(t) = 6.6 + 3.96 t => z''(0) = 6.6 m/s2

So, initial acceleration a = (9.4 i +2.4 j + 6.6k) m/s2

D.)
x(4) = 6.1 + 1.8(4) + 4.7(4)2 + 0.85(4)3   = 142.9 m

y(4) = 8.8 + 8.2(4) + 1.2(4)2 + 0.49(4)3 = 92.16 m

z(4) = 8.3 + 6.4(4) + 3.3(4)2 + 0.66(4)3 = 128.94 m

So, position at t= 4, is (142.9 i + 92.16 j + 128.94 k) m

E.) Velocity at t =4 is,

x' (4) = 1.8 + 9.4(4) + 2.55(4)2 => x'(4) = 80.2 m/s

y'(4)= 8.2 + 2.4 (4) + 1.47(4)2 => y'(4) = 41.32 m/s

z'(4) = 6.4 + 6.6(4) + 1.98(4)2 => z'(4) = 64.48 m/s

So, at t=4 , velocity v = (80.2 i + 41.32 j + 64.48 k) m/s

F.) Acceleration at t=4 seconds is,

x'' (4) = 9.4 + 5.1 (4) => x''(4) = 29.8 m/s2  

y''(4) = 2.4 + 2.94 (4) => y''(4) = 14.16 m/s2

z''(4) = 6.6 + 3.96 (4) => z''(4) = 22.44 m/s2

So, at t=4, acceleration a = (29.8 i +14.16 j + 22.44 k) m/s2

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