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SpyIntel [72]
2 years ago
11

A ball is thrown into the air with initial velocity v(0) = 3i + 8k. The acceleration is given by

Mathematics
1 answer:
Paladinen [302]2 years ago
8 0

Use the fundamental theorem of calculus to find the velocity and position functions.

\displaystyle \mathbf v(t) = \mathbf v(0) + \int_0^t \mathbf a(u) \, du \\\\ ~~~~ = (3\,\mathbf i + 8\,\mathbf k) + (8\,\mathbf j - 16\,\mathbf k) \int_0^t du \\\\ ~~~~ = 3\,\mathbf i + 8t\,\mathbf j + (8-16t)\,\mathbf k

Take the starting position of the ball to be the origin, so \mathbf r(0)=\mathbf 0.

\displaystyle \mathbf r(t) = \mathbf r(0) + \int_0^t \mathbf v(u) \, du \\\\ ~~~~ = \int_0^t \bigg(3\,\mathbf i + 8u\,\mathbf j + (8-16u)\,\mathbf k\bigg) \, du \\\\ ~~~~ = 3t\,\mathbf i + 4t^2\,\mathbf j + (8t - 8t^2)\,\mathbf k

Compute the ball's position vector at t=1.

\mathbf r(1) = 3\,\mathbf i + 4\,\mathbf j

The distance from the ball's starting location at \mathbf r(0) to \mathbf r(1) is then

\|\mathbf r(1) - \mathbf r(0)\| = \|3\,\mathbf i + 4\,\mathbf j\| = \sqrt{3^2 + 4^2} = \boxed{5}

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