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Explain how integration is used to find position from velocity.

Integrating $v(t)$ gives the change in position (displacement). Add the initial position to find the final position.

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Explain how integration is used to find position from velocity.

Integrating $v(t)$ gives the change in position (displacement). Add the initial position to find the final position.

Explain how integration is used to find velocity from acceleration.

Integrating $a(t)$ gives the change in velocity. Add the initial velocity to find the final velocity.

What does the area under the velocity vs. time curve represent?

The area under the $v(t)$ curve represents the displacement of the object.

What does the area under the absolute value of velocity vs. time curve represent?

The area under the $|v(t)|$ curve represents the total distance traveled by the object.

Why is absolute value important when finding distance traveled?

It accounts for changes in direction, ensuring all movement contributes positively to the total distance.

Explain the significance of the constant of integration, C, when integrating velocity or acceleration.

C represents the initial condition (position or velocity) needed to find the specific function.

Describe the relationship between a position vs. time graph and a velocity vs. time graph.

The slope of the position vs. time graph at any point gives the velocity at that time.

Describe the relationship between a velocity vs. time graph and an acceleration vs. time graph.

The slope of the velocity vs. time graph at any point gives the acceleration at that time.

What does a horizontal line on a position vs. time graph indicate?

The object is at rest; its velocity is zero.

What does a horizontal line on a velocity vs. time graph indicate?

The object is moving at a constant velocity; its acceleration is zero.

Formula for velocity given position.

$v(t) = \frac{d}{dt}s(t) = s'(t)$

Formula for acceleration given velocity.

$a(t) = \frac{d}{dt}v(t) = v'(t)$

Formula for acceleration given position.

$a(t) = \frac{d^2}{dt^2}s(t) = s''(t)$

Formula for position given velocity.

$s(t) = \int v(t) dt + C$

Formula for velocity given acceleration.

$v(t) = \int a(t) dt + C$

Formula for displacement.

$\Delta s = s_f - s_i = \int_{t_i}^{t_f} v(t) dt$

Formula for distance traveled.

$\int_{t_i}^{t_f} |v(t)| dt$

How to find final position?

$s(t_f) = s(t_i) + \int_{t_i}^{t_f} v(t) dt$

How to find final velocity?

$v(t_f) = v(t_i) + \int_{t_i}^{t_f} a(t) dt$

What is the relationship between displacement and velocity?

Displacement is the integral of velocity over a time interval.

On a position vs. time graph, what does a steeper slope indicate?

A higher velocity.

On a velocity vs. time graph, what does the area above the x-axis represent?

Positive displacement.

On a velocity vs. time graph, what does the area below the x-axis represent?

Negative displacement.

On a velocity vs. time graph, what does a point on the x-axis represent?

The object is momentarily at rest or changing direction.

On an acceleration vs. time graph, what does the area under the curve represent?

The change in velocity.

How do you find the instantaneous velocity at a specific time from a position vs. time graph?

Find the slope of the tangent line at that time.

How do you find the instantaneous acceleration at a specific time from a velocity vs. time graph?

Find the slope of the tangent line at that time.

What does a concave up position vs. time graph signify?

Positive acceleration; the object is speeding up.

What does a concave down position vs. time graph signify?

Negative acceleration; the object is slowing down.

How can you tell from a position vs. time graph if an object is moving towards or away from the origin?

If the absolute value of the position is increasing, it's moving away; if decreasing, it's moving towards.