Question

a particle is moving with acceleration \\(a(t) = 24t + 6\\). its position at time \\(t = 0\\) is \\(s(0) = 17\\) and its velocity at time \\(t = 0\\) is \\(v(0) = 15\\). what is its position at time \\(t = 14\\)?

Explanation:

Find the velocity function by integration

$$ v(t) = \int a(t) \, dt = \int (24t + 6) \, dt = 12t^2 + 6t + C_1 $$

$$ v(0) = 15 \implies C_1 = 15 \implies v(t) = 12t^2 + 6t + 15 $$

Find the position function by integration

$$ s(t) = \int v(t) \, dt = \int (12t^2 + 6t + 15) \, dt = 4t^3 + 3t^2 + 15t + C_2 $$

$$ s(0) = 17 \implies C_2 = 17 \implies s(t) = 4t^3 + 3t^2 + 15t + 17 $$

Evaluate the position at t = 14

$$ s(14) = 4(14)^3 + 3(14)^2 + 15(14) + 17 $$

$$ s(14) = 4(2744) + 3(196) + 210 + 17 = 10976 + 588 + 210 + 17 = 11791 $$

Answer:

A particle is moving with acceleration \(a(t) = 24t + 6\). its position at time \(t = 0\) is \(s(0) = 17\) and its velocity at time \(t = 0\) is \(v(0) = 15\). What is its position at time \(t = 14\)?
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