Motion Problems, Parametric, Vector-Valued Functions - AP Calculus BC
Card 1 of 30
What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?
What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?
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$\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.
$\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.
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What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?
What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?
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$\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.
$\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.
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Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.
Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.
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$2$. At $t=1$: $\frac{2t}{1} = 2$.
$2$. At $t=1$: $\frac{2t}{1} = 2$.
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Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
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$\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.
$\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.
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Find the position vector at $t=3$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
Find the position vector at $t=3$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
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$\langle 9, 27 \rangle$. Substitute $t=3$: $\langle 9, 27 \rangle$.
$\langle 9, 27 \rangle$. Substitute $t=3$: $\langle 9, 27 \rangle$.
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What is the horizontal component of the velocity vector for $x(t)=5t$?
What is the horizontal component of the velocity vector for $x(t)=5t$?
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$v_x(t) = 5$. Derivative of $5t$ is constant $5$.
$v_x(t) = 5$. Derivative of $5t$ is constant $5$.
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What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?
What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?
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$5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
$5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
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Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.
Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.
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$\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$
$\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$
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What is the horizontal distance traveled by a projectile at $t=5$ for $x(t)=10t$?
What is the horizontal distance traveled by a projectile at $t=5$ for $x(t)=10t$?
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$50$. Substitute $t=5$ into horizontal position.
$50$. Substitute $t=5$ into horizontal position.
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State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.
State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.
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Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.
Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.
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What is the formula for the curvature $\kappa(t)$ of a parametric curve?
What is the formula for the curvature $\kappa(t)$ of a parametric curve?
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$\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.
$\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.
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What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?
What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?
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$\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.
$\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.
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Identify the velocity vector $\mathbf{v}(t)$ given $x(t) = 3t$ and $y(t) = 4t^2$.
Identify the velocity vector $\mathbf{v}(t)$ given $x(t) = 3t$ and $y(t) = 4t^2$.
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$\mathbf{v}(t) = \langle 3, 8t \rangle$. Take derivatives: $v_x = 3$, $v_y = 8t$.
$\mathbf{v}(t) = \langle 3, 8t \rangle$. Take derivatives: $v_x = 3$, $v_y = 8t$.
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What is the formula for total distance traveled in vector form?
What is the formula for total distance traveled in vector form?
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$\int_a^b | \mathbf{v}(t) | , dt$. Integral of speed over time interval.
$\int_a^b | \mathbf{v}(t) | , dt$. Integral of speed over time interval.
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Find the derivative of $x(t) = 6t^3$ with respect to $t$.
Find the derivative of $x(t) = 6t^3$ with respect to $t$.
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$18t^2$. Power rule: derivative of $6t^3$ is $18t^2$.
$18t^2$. Power rule: derivative of $6t^3$ is $18t^2$.
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What is the formula for the binormal vector $\mathbf{B}(t)$ of a curve?
What is the formula for the binormal vector $\mathbf{B}(t)$ of a curve?
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$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$. Cross product of tangent and normal vectors.
$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$. Cross product of tangent and normal vectors.
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State the formula for the tangent vector of a parametric curve at $t = t_0$.
State the formula for the tangent vector of a parametric curve at $t = t_0$.
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$\mathbf{T}(t_0) = \frac{\mathbf{v}(t_0)}{|\mathbf{v}(t_0)|}$. Unit vector formula using velocity magnitude.
$\mathbf{T}(t_0) = \frac{\mathbf{v}(t_0)}{|\mathbf{v}(t_0)|}$. Unit vector formula using velocity magnitude.
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Find the magnitude of the velocity vector $\langle 7, 24 \rangle$.
Find the magnitude of the velocity vector $\langle 7, 24 \rangle$.
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$25$. Magnitude formula: $\sqrt{7^2 + 24^2} = \sqrt{625} = 25$.
$25$. Magnitude formula: $\sqrt{7^2 + 24^2} = \sqrt{625} = 25$.
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State the formula for the position vector $\mathbf{r}(t)$ given $x(t)$ and $y(t)$.
State the formula for the position vector $\mathbf{r}(t)$ given $x(t)$ and $y(t)$.
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$\mathbf{r}(t) = \langle x(t), y(t) \rangle$. Standard vector notation for position.
$\mathbf{r}(t) = \langle x(t), y(t) \rangle$. Standard vector notation for position.
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What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?
What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?
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$5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
$5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
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What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?
What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?
Tap to reveal answer
$\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.
$\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.
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What is the vertical component of the velocity vector for $y(t)=6t^2$?
What is the vertical component of the velocity vector for $y(t)=6t^2$?
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$v_y(t) = 12t$. Derivative of $6t^2$ is $12t$.
$v_y(t) = 12t$. Derivative of $6t^2$ is $12t$.
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Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.
Tap to reveal answer
$\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.
$\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.
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Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.
Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.
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$2$. At $t=1$: $\frac{2t}{1} = 2$.
$2$. At $t=1$: $\frac{2t}{1} = 2$.
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State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.
State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.
Tap to reveal answer
Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.
Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.
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What is the formula for the curvature $\kappa(t)$ of a parametric curve?
What is the formula for the curvature $\kappa(t)$ of a parametric curve?
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$\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.
$\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.
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What is the formula for total distance traveled in vector form?
What is the formula for total distance traveled in vector form?
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$\int_a^b | \mathbf{v}(t) | , dt$. Integral of speed over time interval.
$\int_a^b | \mathbf{v}(t) | , dt$. Integral of speed over time interval.
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What is the parametric equation for $y(t)$ of a projectile launched at angle $\theta$?
What is the parametric equation for $y(t)$ of a projectile launched at angle $\theta$?
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$y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. Vertical motion with gravity acceleration $-g$.
$y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. Vertical motion with gravity acceleration $-g$.
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What is the parametric equation for $x(t)$ of a projectile launched at angle $\theta$?
What is the parametric equation for $x(t)$ of a projectile launched at angle $\theta$?
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$x(t) = v_0 \cos(\theta) t$. Horizontal motion with constant velocity component.
$x(t) = v_0 \cos(\theta) t$. Horizontal motion with constant velocity component.
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Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.
Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.
Tap to reveal answer
$\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$.
$\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$.
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