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SpyIntel [72]
1 year 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]1 year 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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3 years ago
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Fynjy0 [20]
<span>1) Write the point-slope form of the equation of the horizontal line that passes through the point (2, 1). y = 1/2x

2)Write the point-slope form of the equation of the line that passes through the points (6, -9) and (7, 1). 
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3) A line passes through the point (-6, 6) and (-6, 2). In two or more complete sentences, explain why it is not possible to write the equation of the given line in the traditional version of the point-slope form of a line. 

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5) Write the point-slope form of the equation of the line that passes through the points (6, 6) and (-6, 1).
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3 years ago
james is stocking shelves with notebooks. if he has 5 different notebooks, in how many ways can he arrange them on a shelf?
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Step-by-step explanation: I got it right, hope this helps ✨

6 0
3 years ago
Read 2 more answers
the length of a rectangle is 3cm more than the width. find the length and the width if the perimeter of the rectangle is 98cm.
laila [671]

The perimeter of a rectangle is <u>length + length + width + width</u>.

We know that the length of a rectangle is 3cm more than its width, which gives us the equation:  (l for length and w for width)

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We also know that the perimeter of the rectangle is 98cm, which gives us the equation:

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We can divide both sides of this equation by 2 to get:

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Now we'll stick l = 3 + w into the above equation, which gives us:

49 = 3 + w + w

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Now we'll subtract 3 from both sides:

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Dividing both sides by 2 gives us 23 = w.

Substituting w = 23 into the equation l = 3 + w gives us:

l = 3 + 23

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Let's check our answer.  26cm is 3cm more than 23cm.  26cm + 26cm + 23cm + 23cm gives us 98cm.  The length is 26cm and the width is 23cm.


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Answer:

Step-by-step explanation:

This question is incomplete; find the complete question in the attachment.

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y = x²

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y = (x - 2)²

If the curved pit is shifted 2 units up,

Equation of the translated curve will be,

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