AP Calculus BC Flashcards: Motion Problems Parametric Vector Valued Functions

What this deck covers

This deck focuses on Motion Problems Parametric Vector Valued Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the formula for the tangent vector of a parametric curve at $t = t_0$.

Answer: $\mathbf{T}(t_0) = \frac{\mathbf{v}(t_0)}{\|\mathbf{v}(t_0)\|}$. Unit vector formula using velocity magnitude.

Flashcard 2: What is the parametric equation for $x(t)$ of a projectile launched at angle $\theta$?

Answer: $x(t) = v_0 \cos(\theta) t$. Horizontal motion with constant velocity component.

Flashcard 3: What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?

Answer: $5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Flashcard 4: What is the horizontal component of the velocity vector for $x(t)=5t$?

Answer: $v_x(t) = 5$. Derivative of $5t$ is constant $5$.

Flashcard 5: State the formula for the tangent vector of a parametric curve at $t = t_0$.

Answer: $\mathbf{T}(t_0) = \frac{\mathbf{v}(t_0)}{\|\mathbf{v}(t_0)\|}$. Unit vector formula using velocity magnitude.

Flashcard 6: Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

Answer: $\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.

Flashcard 7: What is the vertical component of the velocity vector for $y(t)=6t^2$?

Answer: $v_y(t) = 12t$. Derivative of $6t^2$ is $12t$.

Flashcard 8: What is the formula for total distance traveled in vector form?

Answer: $\int_a^b \| \mathbf{v}(t) \| \, dt$. Integral of speed over time interval.

Flashcard 9: What is the formula for the torsion $\tau(t)$ of a space curve?

Answer: $\tau(t) = -\frac{(\mathbf{B}' \cdot \mathbf{N})}{\|\mathbf{v}\|}$. Standard torsion formula for space curves.

Flashcard 10: What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?

Answer: $\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.

Flashcard 11: What is the parametric equation for $y(t)$ of a projectile launched at angle $\theta$?

Answer: $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. Vertical motion with gravity acceleration $-g$.

Flashcard 12: Find the position vector at $t=3$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

Answer: $\langle 9, 27 \rangle$. Substitute $t=3$: $\langle 9, 27 \rangle$.

Flashcard 13: Identify the position vector $\mathbf{r}(t)$ for $x(t) = 2t$ and $y(t) = 3t^2$.

Answer: $\mathbf{r}(t) = \langle 2t, 3t^2 \rangle$. Combine components into vector form.

Flashcard 14: What is the vertical component of the velocity vector for $y(t)=6t^2$?

Answer: $v_y(t) = 12t$. Derivative of $6t^2$ is $12t$.

Flashcard 15: Find the derivative of $x(t) = 6t^3$ with respect to $t$.

Answer: $18t^2$. Power rule: derivative of $6t^3$ is $18t^2$.

Flashcard 16: What is the formula for the normal vector $\mathbf{N}(t)$ of a curve?

Answer: $\mathbf{N}(t) = \frac{ \frac{d \mathbf{T}}{dt} }{ \left\| \frac{d \mathbf{T}}{dt} \right\| }$. Normalize derivative of unit tangent vector.

Flashcard 17: Find the derivative of $x(t) = 6t^3$ with respect to $t$.

Answer: $18t^2$. Power rule: derivative of $6t^3$ is $18t^2$.

Flashcard 18: State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.

Answer: Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.

Flashcard 19: What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?

Answer: $\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.

Flashcard 20: What is the horizontal distance traveled by a projectile at $t=5$ for $x(t)=10t$?

Answer: $50$. Substitute $t=5$ into horizontal position.

Flashcard 21: Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.

Answer: $\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$

Flashcard 22: Identify the velocity vector $\mathbf{v}(t)$ given $x(t) = 3t$ and $y(t) = 4t^2$.

Answer: $\mathbf{v}(t) = \langle 3, 8t \rangle$. Take derivatives: $v_x = 3$, $v_y = 8t$.

Flashcard 23: State the formula for speed of a particle given $v_x(t)$ and $v_y(t)$.

Answer: Speed = $\sqrt{v_x(t)^2 + v_y(t)^2}$. Pythagorean theorem for vector magnitude.

Flashcard 24: Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.

Answer: $2$. At $t=1$: $\frac{2t}{1} = 2$.

Flashcard 25: What is the formula for the binormal vector $\mathbf{B}(t)$ of a curve?

Answer: $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$. Cross product of tangent and normal vectors.

Flashcard 26: Identify the velocity vector $\mathbf{v}(t)$ given $x(t) = 3t$ and $y(t) = 4t^2$.

Answer: $\mathbf{v}(t) = \langle 3, 8t \rangle$. Take derivatives: $v_x = 3$, $v_y = 8t$.

Flashcard 27: Find the derivative of $y(t) = 5t^2$ with respect to $t$.

Answer: $10t$. Power rule: derivative of $5t^2$ is $10t$.

Flashcard 28: Identify the position vector $\mathbf{r}(t)$ for $x(t) = 2t$ and $y(t) = 3t^2$.

Answer: $\mathbf{r}(t) = \langle 2t, 3t^2 \rangle$. Combine components into vector form.

Flashcard 29: State the formula for the position vector $\mathbf{r}(t)$ given $x(t)$ and $y(t)$.

Answer: $\mathbf{r}(t) = \langle x(t), y(t) \rangle$. Standard vector notation for position.

Flashcard 30: Find the value of $\frac{dy}{dx}$ for $y(t)=t^2$, $x(t)=t+1$ at $t=1$.

Answer: $2$. At $t=1$: $\frac{2t}{1} = 2$.

Flashcard 31: Find $\frac{dy}{dx}$ for $y(t)=t^3$, $x(t)=t^2$ at $t=1$.

Answer: $\frac{3}{2}$. Chain rule: $\frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{3t^2}{2t} = \frac{3}{2}$.

Flashcard 32: State the formula for arc length of a parametric curve from $t=a$ to $t=b$.

Answer: $\int_a^b \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. Integral of speed over time interval.

Flashcard 33: What is the formula for the curvature $\kappa(t)$ of a parametric curve?

Answer: $\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.

Flashcard 34: What is the formula for the normal vector $\mathbf{N}(t)$ of a curve?

Answer: $\mathbf{N}(t) = \frac{ \frac{d \mathbf{T}}{dt} }{ \| \frac{d \mathbf{T}}{dt} \| }$. Normalize derivative of unit tangent vector.

Flashcard 35: What is the formula for total distance traveled in vector form?

Answer: $\int_a^b \| \mathbf{v}(t) \| \, dt$. Integral of speed over time interval.

Flashcard 36: What is the formula for the binormal vector $\mathbf{B}(t)$ of a curve?

Answer: $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$. Cross product of tangent and normal vectors.

Flashcard 37: Find the magnitude of the velocity vector $\langle 7, 24 \rangle$.

Answer: $25$. Magnitude formula: $\sqrt{7^2 + 24^2} = \sqrt{625} = 25$.

Flashcard 38: What is the parametric equation for $y(t)$ of a projectile launched at angle $\theta$?

Answer: $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$. Vertical motion with gravity acceleration $-g$.

Flashcard 39: Find the position vector at $t=3$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

Answer: $\langle 9, 27 \rangle$. Substitute $t=3$: $\langle 9, 27 \rangle$.

Flashcard 40: What is the horizontal component of the velocity vector for $x(t)=5t$?

Answer: $v_x(t) = 5$. Derivative of $5t$ is constant $5$.

Flashcard 41: State the formula for the position vector $\mathbf{r}(t)$ given $x(t)$ and $y(t)$.

Answer: $\mathbf{r}(t) = \langle x(t), y(t) \rangle$. Standard vector notation for position.

Flashcard 42: Find the unit tangent vector at $t=1$ for $\mathbf{r}(t) = \langle t^2, t^3 \rangle$.

Answer: $\langle \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \rangle$. Normalize $\langle 2, 3 \rangle$ with magnitude $\sqrt{13}$.

Flashcard 43: What is the velocity vector $\mathbf{v}(t)$ for $x(t) = t^2$, $y(t) = t^3$?

Answer: $\langle 2t, 3t^2 \rangle$. Derivatives: $v_x = 2t$, $v_y = 3t^2$.

Flashcard 44: State the formula for arc length of a parametric curve from $t=a$ to $t=b$.

Answer: $\int_a^b \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. Integral of speed over time interval.

Flashcard 45: What is the magnitude of the acceleration vector $\mathbf{a}(t) = \langle 3, 4 \rangle$?

Answer: $5$. Magnitude formula: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Flashcard 46: Find the derivative of $y(t) = 5t^2$ with respect to $t$.

Answer: $10t$. Power rule: derivative of $5t^2$ is $10t$.

Flashcard 47: What is the formula for the curvature $\kappa(t)$ of a parametric curve?

Answer: $\kappa(t) = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}$. Standard curvature formula for parametric curves.

Flashcard 48: Find the magnitude of the velocity vector $\langle 7, 24 \rangle$.

Answer: $25$. Magnitude formula: $\sqrt{7^2 + 24^2} = \sqrt{625} = 25$.

Flashcard 49: What is the formula for the torsion $\tau(t)$ of a space curve?

Answer: $\tau(t) = -\frac{(\mathbf{B}' \cdot \mathbf{N})}{\|\mathbf{v}\|}$. Standard torsion formula for space curves.

Flashcard 50: What is the parametric equation for $x(t)$ of a projectile launched at angle $\theta$?

Answer: $x(t) = v_0 \cos(\theta) t$. Horizontal motion with constant velocity component.

Flashcard 51: What is the horizontal distance traveled by a projectile at $t=5$ for $x(t)=10t$?

Answer: $50$. Substitute $t=5$ into horizontal position.

Flashcard 52: What is the acceleration vector $\mathbf{a}(t)$ for $x(t) = 4t^2$ and $y(t) = 2t^3$?

Answer: $\mathbf{a}(t) = \langle 8, 12t \rangle$. Second derivatives: $a_x = 8$, $a_y = 12t$.