Connecting Position, Velocity, and Acceleration - AP Calculus AB
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Find the position at $t = 4$ if $v(t) = 3t$ and $s(0) = 2$.
Find the position at $t = 4$ if $v(t) = 3t$ and $s(0) = 2$.
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$s(4) = 26$. $s(t) = \int 3t , dt = \frac{3t^2}{2} + C$, with $s(0) = 2$ gives $s(4) = 26$.
$s(4) = 26$. $s(t) = \int 3t , dt = \frac{3t^2}{2} + C$, with $s(0) = 2$ gives $s(4) = 26$.
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What is the formula for speed given velocity $v(t)$?
What is the formula for speed given velocity $v(t)$?
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Speed = $|v(t)|$. Speed is the magnitude (absolute value) of velocity.
Speed = $|v(t)|$. Speed is the magnitude (absolute value) of velocity.
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Identify the derivative representing instantaneous velocity.
Identify the derivative representing instantaneous velocity.
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$v(t) = s'(t)$. First derivative of position gives instantaneous velocity.
$v(t) = s'(t)$. First derivative of position gives instantaneous velocity.
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What is the formula for velocity given position function $s(t)$?
What is the formula for velocity given position function $s(t)$?
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$v(t) = s'(t)$. Velocity is the rate of change of position with respect to time.
$v(t) = s'(t)$. Velocity is the rate of change of position with respect to time.
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What is the interpretation of zero acceleration?
What is the interpretation of zero acceleration?
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The velocity is constant. No change in velocity means constant speed and direction.
The velocity is constant. No change in velocity means constant speed and direction.
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What mathematical concept describes the rate of change of velocity?
What mathematical concept describes the rate of change of velocity?
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Acceleration. Acceleration measures how quickly velocity changes.
Acceleration. Acceleration measures how quickly velocity changes.
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What condition indicates a change in direction of motion?
What condition indicates a change in direction of motion?
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Velocity changes sign. Sign change in velocity indicates direction reversal.
Velocity changes sign. Sign change in velocity indicates direction reversal.
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What is the graphical representation of velocity?
What is the graphical representation of velocity?
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Slope of the tangent line on position-time graph. Velocity equals the slope of the position curve at any point.
Slope of the tangent line on position-time graph. Velocity equals the slope of the position curve at any point.
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What is the formula for acceleration given velocity function $v(t)$?
What is the formula for acceleration given velocity function $v(t)$?
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$a(t) = v'(t)$. Acceleration is the rate of change of velocity with respect to time.
$a(t) = v'(t)$. Acceleration is the rate of change of velocity with respect to time.
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State the sign of acceleration when velocity is increasing.
State the sign of acceleration when velocity is increasing.
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Acceleration is positive. Positive acceleration means velocity is increasing over time.
Acceleration is positive. Positive acceleration means velocity is increasing over time.
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What is the significance of a velocity-time graph crossing the time axis?
What is the significance of a velocity-time graph crossing the time axis?
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Change in direction of motion. Crossing indicates velocity changes sign, reversing direction.
Change in direction of motion. Crossing indicates velocity changes sign, reversing direction.
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Find the velocity at $t = 3$ for $s(t) = 2t^2 - 3t + 4$.
Find the velocity at $t = 3$ for $s(t) = 2t^2 - 3t + 4$.
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$v(3) = 9$. $v(t) = s'(t) = 4t - 3$, so $v(3) = 12 - 3 = 9$.
$v(3) = 9$. $v(t) = s'(t) = 4t - 3$, so $v(3) = 12 - 3 = 9$.
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What does a constant velocity imply about acceleration?
What does a constant velocity imply about acceleration?
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Acceleration is zero. Constant velocity means no change in velocity, so $a = 0$.
Acceleration is zero. Constant velocity means no change in velocity, so $a = 0$.
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What is the integral of acceleration $a(t)$ with respect to $t$?
What is the integral of acceleration $a(t)$ with respect to $t$?
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Velocity $v(t)$. Integrating acceleration gives velocity function.
Velocity $v(t)$. Integrating acceleration gives velocity function.
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State the formula for average velocity over time interval $[a, b]$.
State the formula for average velocity over time interval $[a, b]$.
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$v_{avg} = \frac{s(b) - s(a)}{b - a}$. Change in position divided by change in time.
$v_{avg} = \frac{s(b) - s(a)}{b - a}$. Change in position divided by change in time.
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Find the acceleration at $t = 2$ for $v(t) = 3t^2 + 2t$.
Find the acceleration at $t = 2$ for $v(t) = 3t^2 + 2t$.
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$a(2) = 14$. $a(t) = v'(t) = 6t + 2$, so $a(2) = 12 + 2 = 14$.
$a(2) = 14$. $a(t) = v'(t) = 6t + 2$, so $a(2) = 12 + 2 = 14$.
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Find the velocity function given $a(t) = 6$ and $v(0) = 4$.
Find the velocity function given $a(t) = 6$ and $v(0) = 4$.
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$v(t) = 6t + 4$. $v(t) = \int 6 , dt = 6t + C$, with $v(0) = 4$.
$v(t) = 6t + 4$. $v(t) = \int 6 , dt = 6t + C$, with $v(0) = 4$.
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Find the change in velocity over $[0, 3]$ for $a(t) = 4t$.
Find the change in velocity over $[0, 3]$ for $a(t) = 4t$.
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$\text{Change} = 18$. $\int_0^3 4t , dt = 2t^2 |_0^3 = 18$.
$\text{Change} = 18$. $\int_0^3 4t , dt = 2t^2 |_0^3 = 18$.
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What is the relationship between velocity and acceleration?
What is the relationship between velocity and acceleration?
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Acceleration is the derivative of velocity. Rate of change of velocity gives instantaneous acceleration.
Acceleration is the derivative of velocity. Rate of change of velocity gives instantaneous acceleration.
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Find the velocity when $a(t) = 3t$ and $v(0) = 2$.
Find the velocity when $a(t) = 3t$ and $v(0) = 2$.
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$v(t) = \frac{3}{2}t^2 + 2$. $v(t) = \int 3t , dt = \frac{3t^2}{2} + C$, with $v(0) = 2$.
$v(t) = \frac{3}{2}t^2 + 2$. $v(t) = \int 3t , dt = \frac{3t^2}{2} + C$, with $v(0) = 2$.
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What is the relationship between position and velocity?
What is the relationship between position and velocity?
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Velocity is the derivative of position. Rate of change of position gives instantaneous velocity.
Velocity is the derivative of position. Rate of change of position gives instantaneous velocity.
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What is the effect of zero velocity over a time interval?
What is the effect of zero velocity over a time interval?
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No displacement occurs. Zero velocity means no change in position occurs.
No displacement occurs. Zero velocity means no change in position occurs.
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If $s(t) = 4t^3 - 3t^2$, find the velocity function $v(t)$.
If $s(t) = 4t^3 - 3t^2$, find the velocity function $v(t)$.
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$v(t) = 12t^2 - 6t$. Velocity is the first derivative of position function.
$v(t) = 12t^2 - 6t$. Velocity is the first derivative of position function.
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What does a velocity graph's horizontal tangent indicate?
What does a velocity graph's horizontal tangent indicate?
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Zero acceleration. Horizontal tangent means zero slope, so zero acceleration.
Zero acceleration. Horizontal tangent means zero slope, so zero acceleration.
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What does it mean if velocity and acceleration have opposite signs?
What does it mean if velocity and acceleration have opposite signs?
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The object is slowing down. Opposite signs indicate velocity and acceleration work against each other.
The object is slowing down. Opposite signs indicate velocity and acceleration work against each other.
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What is the derivative of velocity?
What is the derivative of velocity?
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Acceleration. First derivative of velocity equals acceleration.
Acceleration. First derivative of velocity equals acceleration.
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What is the second derivative of position $s(t)$?
What is the second derivative of position $s(t)$?
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Acceleration $a(t) = s''(t)$. Second derivative of position gives acceleration function.
Acceleration $a(t) = s''(t)$. Second derivative of position gives acceleration function.
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Calculate $s(t)$ given $v(t) = 2t + 1$ and $s(0) = 3$.
Calculate $s(t)$ given $v(t) = 2t + 1$ and $s(0) = 3$.
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$s(t) = t^2 + t + 3$. $s(t) = \int (2t + 1) , dt = t^2 + t + C$, with $s(0) = 3$.
$s(t) = t^2 + t + 3$. $s(t) = \int (2t + 1) , dt = t^2 + t + C$, with $s(0) = 3$.
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What is the effect of constant positive acceleration?
What is the effect of constant positive acceleration?
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Velocity increases linearly. Constant acceleration produces linear increase in velocity.
Velocity increases linearly. Constant acceleration produces linear increase in velocity.
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What does a zero velocity at an instant imply about motion?
What does a zero velocity at an instant imply about motion?
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Object is momentarily at rest. Zero velocity indicates no motion at that instant.
Object is momentarily at rest. Zero velocity indicates no motion at that instant.
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