Frame $A$ moves with constant velocity $v_0$ relative to the ground along a straight line. At time $t = 0$, the origins of frame $A$ and frame $B$ coincide at $x = 0$, and frame $B$ is at rest relative to frame $A$. Frame $B$ accelerates relative to frame $A$ with acceleration $a_{B/A}(t) = bt$, where $b$ is a positive constant. Which of the following expressions represents the position of the origin of frame $B$ relative to the ground, $x_{B/G}(t)$, as a function of time $t$? • Nerd Notes

AP Physics

Unit 1 - Vectors and Kinematics

MCQ

Mathematical

Intermediate

Step	Reasoning
Identify the target equation for relative position. \[ x_{B/G}(t) = x_{B/A}(t) + x_{A/G}(t) \]	The question asks for the position of frame $B$ relative to the ground, $x_{B/G}(t)$, which can be found by adding the position of frame $B$ relative to frame $A$ to the position of frame $A$ relative to the ground.
Determine the position of frame $A$ relative to the ground. \[ x_{A/G}(t) = v_0 t \]	Frame $A$ moves at a constant velocity $v_0$ relative to the ground starting from $x = 0$ at $t = 0$.
Integrate relative acceleration to find relative velocity $v_{B/A}(t)$. \[ v_{B/A}(t) = \int_0^t bt’\,dt’ = \dfrac{1}{2}b t^2 \]	To find $x_{B/A}(t)$, we must first integrate $a_{B/A}(t) = bt$ with initial condition $v_{B/A}(0) = 0$.
Integrate relative velocity to find relative position $x_{B/A}(t)$. \[ x_{B/A}(t) = \int_0^t \dfrac{1}{2}b (t’)^2\,dt’ = \dfrac{1}{6}b t^3 \]	Integrating $v_{B/A}(t)$ with respect to time with initial condition $x_{B/A}(0) = 0$ gives the position of frame $B$ relative to frame $A$.
Combine terms to solve for $x_{B/G}(t)$. \[ x_{B/G}(t) = v_0 t + \dfrac{1}{6}b t^3 \]	Substitute $x_{B/A}(t)$ and $x_{A/G}(t)$ back into the relative position equation.

Why each choice is correct or incorrect:

(A) This choice integrates acceleration only once, yielding velocity instead of position, or incorrectly assumes constant acceleration.

(B) This choice omits the factor of $\dfrac{1}{2}$ during the first integration step, leading to an incorrect coefficient of $\dfrac{1}{3}$.

(C) This choice correctly computes $v_{B/A}(t) = \dfrac{1}{2}bt^2$ but drops the factor of $\dfrac{1}{3}$ during the second integration to position.

(D) This is the correct answer obtained by integrating $a_{B/A}(t) = bt$ twice and adding the relative motion of frame $A$.

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A conducting loop is fixed in a region with a magnetic field directed perpendicular to the plane of the loop. The induced electromotive force (EMF) $\mathcal{E}$ in the loop is measured as a function of time $t$ and is shown in the graph. At time $t = 0$, the magnetic flux through the loop is zero ($\Phi_B = 0$). Which of the following graphs best represents the magnetic flux $\Phi_B$ through the loop as a function of time $t$?

Intermediate

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A rigid circular wire loop of radius $a$ and total resistance $R$ lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time $t$ according to $B(t) = B_0 \left(1 + \dfrac{t}{\tau}\right)^3$, where $B_0$ and $\tau$ are positive constants. Which of the following expressions represents the total electric charge $Q$ that flows past a point in the loop between $t = 0$ and $t = \tau$?

Advanced

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MCQ

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Beginner

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MCQ

An ideal circuit consists of an inductor of inductance $L$ connected in series with a capacitor of capacitance $C$. At time $t = 0$, the capacitor carries an initial charge $Q_0$ and the current in the circuit is zero. Which of the following graphs best represents the charge $q(t)$ on the capacitor as a function of time $t$?

Intermediate

Mathematical

MCQ

$A 3D Cartesian coordinate system with x, y, and z axes labeled at the ends of thin black lines with arrowheads. A flat square loop of side length L is situated in the xy-plane with one edge lying along the y-axis from y = 0 to y = L. A second identical square loop shares the same edge along the y-axis but is tilted upward into the xz-plane by an angle \theta relative to the xy-plane. An arc labeled \theta indicates the tilt angle between the two loop planes. Vertical upward arrows parallel to the z-axis represent a magnetic field \vec{B}; the arrows increase in length at larger values of x. No other labels, lines, text, or axes appear.$

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Advanced

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A horizontal circular ring of radius $r$, mass $m$, and electrical resistance $R$ falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height $z$ according to $B_z(z) = B_0 + bz$, where $B_0$ and $b$ are positive constants, and $z$ is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring’s terminal velocity $v_T$?

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$A circular conducting disk of radius R lies flat in the xy-plane, centered at the origin. A vertical z-axis extends upward through the center of the disk, with a curved arrow around the z-axis indicating rotation at constant angular velocity \omega. A radial line segment extends from the origin to the outer rim of the disk, labeled R. Several upward-pointing magnetic field vectors \vec{B}(r) pass vertically through the disk, with shorter vectors near the center and longer vectors near the outer rim to represent the radial increase in field magnitude. No other labels, lines, text, or axes appear.$

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Advanced

Mathematical

MCQ

A long, straight cylindrical wire of radius $R$ carries a steady total current $I$ distributed uniformly across its circular cross section. The permeability of free space is $\mu_0$. Which of the following expressions represents the total magnetic energy stored per unit length inside the volume of the wire?

Advanced

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MCQ

A circular region of radius $R$ in the $xy$-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region ($r \le R$), the magnitude of the magnetic field is given by $B(r,t) = C t \left(\dfrac{r}{R}\right)^2$, where $C$ is a positive constant and $r$ is the radial distance from the central axis. Outside the region ($r > R$), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field $E(r)$ as a function of distance $r$ from the central axis inside the region ($r < R$)?

Advanced

Mathematical

MCQ

$A schematic diagram of an electric circuit drawn with straight black lines forming two adjacent rectangular loops. On the far left vertical wire of the left loop is a DC battery labeled $V_0$ with its longer positive terminal on top. On the top horizontal wire of the left loop is a switch labeled $S$ shown in an open position and a resistor labeled $R_1 = R$ in series. A shared vertical wire separates the left and right loops, containing a resistor labeled $R_2 = 2R$. On the far right vertical wire of the right loop, a resistor labeled $R_3 = R$ is connected in series above an inductor coil labeled $L$. No other labels, lines, text, or axes appear.$

In the circuit shown, an ideal battery with potential difference $V_0$ is connected to three resistors ($R_1 = R$, $R_2 = 2R$, $R_3 = R$), an ideal inductor $L$, and a switch $S$. The switch $S$ has been closed for a long time. At time $t = 0$, switch $S$ is opened. Which of the following expressions represents the magnitude of the current $I(t)$ through resistor $R_2$ as a function of time $t$ for $t \ge 0$?

Step	Reasoning
Identify the target equation for relative position. \[ x_{B/G}(t) = x_{B/A}(t) + x_{A/G}(t) \]	The question asks for the position of frame \(B\) relative to the ground, \(x_{B/G}(t)\), which can be found by adding the position of frame \(B\) relative to frame \(A\) to the position of frame \(A\) relative to the ground.
Determine the position of frame \(A\) relative to the ground. \[ x_{A/G}(t) = v_0 t \]	Frame \(A\) moves at a constant velocity \(v_0\) relative to the ground starting from \(x = 0\) at \(t = 0\).
Integrate relative acceleration to find relative velocity \(v_{B/A}(t)\). \[ v_{B/A}(t) = \int_0^t bt’\,dt’ = \dfrac{1}{2}b t^2 \]	To find \(x_{B/A}(t)\), we must first integrate \(a_{B/A}(t) = bt\) with initial condition \(v_{B/A}(0) = 0\).
Integrate relative velocity to find relative position \(x_{B/A}(t)\). \[ x_{B/A}(t) = \int_0^t \dfrac{1}{2}b (t’)^2\,dt’ = \dfrac{1}{6}b t^3 \]	Integrating \(v_{B/A}(t)\) with respect to time with initial condition \(x_{B/A}(0) = 0\) gives the position of frame \(B\) relative to frame \(A\).
Combine terms to solve for \(x_{B/G}(t)\). \[ x_{B/G}(t) = v_0 t + \dfrac{1}{6}b t^3 \]	Substitute \(x_{B/A}(t)\) and \(x_{A/G}(t)\) back into the relative position equation.

Kinematics	Forces
\(\Delta x = v_i t + \frac{1}{2} at^2\)	\(F = ma\)
\(v = v_i + at\)	\(F_g = \frac{G m_1 m_2}{r^2}\)
\(v^2 = v_i^2 + 2a \Delta x\)	\(f = \mu N\)
\(\Delta x = \frac{v_i + v}{2} t\)	\(F_s =-kx\)
\(v^2 = v_f^2 \,-\, 2a \Delta x\)

Circular Motion	Energy
\(F_c = \frac{mv^2}{r}\)	\(KE = \frac{1}{2} mv^2\)
\(a_c = \frac{v^2}{r}\)	\(PE = mgh\)
\(T = 2\pi \sqrt{\frac{r}{g}}\)	\(KE_i + PE_i = KE_f + PE_f\)
	\(W = Fd \cos\theta\)

Momentum	Torque and Rotations
\(p = mv\)	\(\tau = r \cdot F \cdot \sin(\theta)\)
\(J = \Delta p\)	\(I = \sum mr^2\)
\(p_i = p_f\)	\(L = I \cdot \omega\)

Simple Harmonic Motion	Fluids
\(F = -kx\)	\(P = \frac{F}{A}\)
\(T = 2\pi \sqrt{\frac{l}{g}}\)	\(P_{\text{total}} = P_{\text{atm}} + \rho gh\)
\(T = 2\pi \sqrt{\frac{m}{k}}\)	\(Q = Av\)
\(x(t) = A \cos(\omega t + \phi)\)	\(F_b = \rho V g\)
\(a = -\omega^2 x\)	\(A_1v_1 = A_2v_2\)

Constant	Description
[katex]g[/katex]	Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex]	Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s[/katex]	Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex]	Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex]	Mass of the Earth
[katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex]	Mass of the Moon
[katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex]	Mass of the Sun

Variable	SI Unit
[katex]s[/katex] (Displacement)	[katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity)	[katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration)	[katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time)	[katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass)	[katex]\text{kilograms (kg)}[/katex]

Variable	Derived SI Unit
[katex]F[/katex] (Force)	[katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy)	[katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power)	[katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum)	[katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\omega[/katex] (Angular Velocity)	[katex]\text{radians per second (rad/s)}[/katex]
[katex]\tau[/katex] (Torque)	[katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia)	[katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency)	[katex]\text{hertz (Hz)}[/katex]

Prefix	Symbol	Power of Ten
Pico-	p	[katex]10^{-12}[/katex]
Nano-	n	[katex]10^{-9}[/katex]
Micro-	µ	[katex]10^{-6}[/katex]
Milli-	m	[katex]10^{-3}[/katex]
Centi-	c	[katex]10^{-2}[/katex]
Deci-	d	[katex]10^{-1}[/katex]
(Base unit)	–	[katex]10^{0}[/katex]
Deca- or Deka-	da	[katex]10^{1}[/katex]
Hecto-	h	[katex]10^{2}[/katex]
Kilo-	k	[katex]10^{3}[/katex]
Mega-	M	[katex]10^{6}[/katex]
Giga-	G	[katex]10^{9}[/katex]
Tera-	T	[katex]10^{12}[/katex]

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AP Physics

Unit 1 - Vectors and Kinematics

MCQ

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Intermediate

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