Position, Velocity, and Acceleration using Integrals Practice

Sample questions

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1. Sample 1difficulty 2/5

   Average velocity of $v(t) = t^2 - 2t$ on $[0, 3]$ is

   • A

     $\tfrac{3}{2}$

   • B

     $0$

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   • C

     $1$

   • D

     $3$

   Why

   $\dfrac{1}{3}\int_0^3 (t^2 - 2t)\,dt = (1/3)(9 - 9) = 0$.

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2. Sample 2difficulty 2/5

   A particle moves with $v(t) = \cos t$ on $[0, 2\pi]$. Total displacement:

   • A

     $4$

   • B

     $2\pi$

   • C

     $2$

   • D

     $0$

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   Why

   $\int_0^{2\pi}\cos t\,dt = 0$.

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3. Sample 3difficulty 2/5

   For a particle with $v(t) = 2t$, the displacement over $[0, 3]$ is

   • A

     $18$

   • B

     $9$

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   • C

     $6$

   • D

     $3$

   Why

   $\int_0^3 2t\,dt = [t^2]_0^3 = 9$.

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4. Sample 4difficulty 2/5

   With $v(t) = 2t + 1$ and $s(0) = 4$, find $s(3)$.

   • A

     $13$

   • B

     $16$

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   • C

     $10$

   • D

     $22$

   Why

   $s(t) = t^2 + t + C$. $s(0) = 4 = C$. $s(3) = 9 + 3 + 4 = 16$.

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5. Sample 5difficulty 2/5

   A particle starts at $s(0) = 5$ with velocity $v(t) = 3t$. Position at $t = 4$:

   • A

     $24$

   • B

     $29$

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   • C

     $48$

   • D

     $12$

   Why

   $s(t) = 5 + \int_0^t 3u\,du = 5 + 3t^2/2$. $s(4) = 5 + 24 = 29$.

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