Motion in a Straight Line - Class 11 Part 1

A closed container is completely filled with a viscous incompressible fluid of density $\rho_f$. The container accelerates horizontally along the positive x-axis with a constant acceleration $\vec{a} = g\hat{i}$, where $g$ is the acceleration due to gravity (acting along $-g\hat{j}$). A small spherical particle of mass $m$, charge $+q$, and density $\rho_p = 2\rho_f$ moves in the fluid. A uniform magnetic field $\vec{B} = B_0\hat{k}$ is present in the region. The fluid exerts a drag force $\vec{F}_D = -b\vec{v}$, where $\vec{v}$ is the particle's velocity relative to the container. If the drag coefficient is $b = qB_0$, what is the steady-state terminal velocity $\vec{v}_T$ of the particle relative to the container?

A thin converging lens of focal length $f$ (in air) is placed at the origin $x = 0$. The region $x < 0$ is filled with air (refractive index $\mu = 1$). The region $x > 0$ is filled with a transparent medium whose refractive index varies along the x-axis as $\mu(x) = 1 + \alpha x$, where $\alpha > 0$ is a constant. Let $F_1$ be the first focal point (the object position on the principal axis for which the image is formed at infinity) and $F_2$ be the second focal point (the image position for an object at infinity). If a point object is placed at a distance $x_1$ to the left of $F_1$, its image is formed at a distance $x_2$ to the right of $F_2$. Which of the following represents the modified Newton's lens equation for this system?

A primary coil is driven by an alternating current $I_1(t) = I_0 \sin(\omega t)$. A secondary coil with self-inductance $L$ and negligible resistance is magnetically coupled to the primary with a mutual inductance $M > 0$. The secondary coil is connected in series with a load $Z$. The reference direction for the secondary current $I_2(t)$ is chosen such that positive $I_1$ and positive $I_2$ produce magnetic flux in the same direction in the core. Let $\Delta \phi$ be the phase difference such that the steady-state secondary current is $I_2(t) = I_{20} \sin(\omega t + \Delta \phi)$ with $I_{20} > 0$. If $\Delta \phi_L$ is the phase difference when $Z$ is an ideal inductor $L_0$, and $\Delta \phi_C$ is the phase difference when $Z$ is an ideal capacitor $C_0$ (where $\frac{1}{\omega C_0} > \omega L$), what are the values of $\Delta \phi_L$ and $\Delta \phi_C$ respectively?

A massive string of length $L$ and total mass $M$ has a non-uniform linear mass density $\lambda(x) = kx$, where $x$ is the distance from the fixed pivot. The string is given an initial angular velocity at its lowest position ($\theta = 0$) such that the tension at the pivot is exactly zero when the string reaches the highest point ($\theta = \pi$). Find the tension in the string at its midpoint ($x = L/2$) when it is at an angle $\theta = 2\pi/3$ from the downward vertical.

A particle starts from rest at the origin $x=0$ at time $t=0$. It is subjected to a time-varying force such that its acceleration $a(t)$ is directly proportional to the instantaneous activity $A(t)$ of a radioactive sample. The sample has an initial number of active nuclei $N_0$ and a decay constant $\lambda$. Given the proportionality constant is $k$ (such that $a(t) = k A(t)$), what is the displacement of the particle at the exact moment the activity of the sample reduces to $\frac{1}{e}$ of its initial value?

A solid cylinder of mass $m$ and radius $R$ is projected with an initial translational velocity $v_0$ and zero angular velocity on a rough horizontal surface with a coefficient of kinetic friction $\mu_k$. The cylinder experiences slipping before transitioning to pure rolling. Let $A_v$ be the area under the velocity-time ($v-t$) graph and $A_\omega$ be the area under the angular velocity-time ($\omega-t$) graph from $t=0$ until the exact moment pure rolling commences. Which of the following expressions correctly represents the total mechanical energy dissipated due to friction during this process?

A particle of mass $m = 1 \text{ kg}$ is projected from the origin $x=0$ along the positive x-axis with an initial speed $v_0 = 20 \text{ m/s}$ on a smooth horizontal surface. The particle moves through a resistive medium that exerts a drag force of magnitude $F_d = kS$, where $k = 4 \text{ N/m}$ and $S$ is the total path length covered by the particle from the start of its motion. The force always acts opposite to the velocity vector. At $x = L = 2 \text{ m}$, the particle undergoes a head-on inelastic collision with a fixed vertical wall. If the coefficient of restitution is $e = 0.5$, what is the kinetic energy of the particle just as it returns to the origin $x=0$?

Initially at $t=0$, a system comprising a large parallel-plate capacitor and an uncharged, perfectly conducting solid block placed entirely between its plates is completely field-free. The block contains an internal, empty, eccentric spherical cavity of radius $r_0$. For $t > 0$, the capacitor is charged such that a uniform displacement current density $\vec{J}_d = J_0 \hat{k}$ exists in the vacuum region between the plates. An infinitesimal test charge $+q_0$ is moved from the center of the cavity to its inner surface along a complex path of total length $L$ at a constant speed $v$. What is the magnitude of the total work done by the electromagnetic forces on the test charge during this process?

Two parallel ideal conducting rails are placed at $y = a$ and $y = -a$ in the $xy$-plane, extending from $x=0$ to $x \to \infty$. They are connected at $x=0$ by a zero-resistance wire. A non-uniform magnetic field $\vec{B} = B_0 \frac{x}{a} \hat{k}$ exists in the region $x > 0$. Two rigid conducting wires, $W_1$ and $W_2$, each with uniform resistance $\rho$ per unit length, are placed across the rails. $W_1$ is a straight wire connecting the points $(x_0, a)$ and $(x_0, -a)$. $W_2$ is a semicircular wire bulging in the $+x$ direction, with its ends at $(x_0, a)$ and $(x_0, -a)$. Both wires are pulled independently in the $+x$ direction with a constant velocity $\vec{v} = v_0 \hat{i}$. At the instant $x_0 = a$, let $P_1$ and $P_2$ be the instantaneous thermal powers dissipated in $W_1$ and $W_2$ respectively. Neglecting self-inductance, what is the ratio $\frac{P_1}{P_2}$?

A block of mass $m = 1 \text{ kg}$ is initially at rest on a rough horizontal surface. The coefficients of static and kinetic friction are both $\mu = 0.2$. At $t = 0$, a time-varying horizontal force $F(t) = 10 - 4t \text{ N}$ is applied to the block. Take $g = 10 \text{ m/s}^2$. Let $W$ be the magnitude of the total work done by the frictional force on the block, and $J$ be the magnitude of the total impulse delivered by the frictional force, both evaluated from $t = 0$ to $t = 5 \text{ s}$. What is the value of the ratio $\frac{W}{J}$?

A particle of mass $m = 2 \text{ kg}$ is constrained to move along the $x$-axis under the influence of a conservative force field. The potential energy of the particle is given by $U(x) = \left(-\frac{1}{3}x^3 + 2x^2 - 3x\right) \text{ J}$, where $x$ is the position in meters. If the particle is released from rest at the origin ($x = 0$), what is its speed when its acceleration becomes zero for the first time?

A particle of mass $m$ is driven along a straight horizontal axis by an engine that delivers a constant power $P$. The particle experiences a resistive drag force of magnitude $k v^2$, where $v$ is the instantaneous velocity and $k$ is a positive constant. If the particle starts from rest at the origin, what is the distance $x$ covered by the particle when it attains a velocity $v_0$ (where $v_0 < (P/k)^{1/3}$)?

An alpha particle of mass $m$ is fired directly towards a heavy stationary nucleus. The repulsive Coulomb force on the alpha particle is given by $F = \frac{k}{x^2}$, where $x$ is the distance from the nucleus and $k$ is a positive constant. The particle passes the point $x = 3d$ with a velocity $v_0$ directed towards the nucleus. Assuming $v_0 \ll c$ (where $c$ is the speed of light), what is the square of its velocity, $v^2$, when it reaches the point $x = d$?

A particle of mass $m$ (in $\text{kg}$) is constrained to move along the $x$-axis. It is subjected to a restoring force $F = -kx - \alpha x^3$, where $k$ (in $\text{N m}^{-1}$) and $\alpha$ (in $\text{N m}^{-3}$) are positive constants. If the particle is released from rest at position $x = A$ (in $\text{m}$), what is its speed $v$ (in $\text{m s}^{-1}$) when it passes through the origin ($x = 0$)?

A block of mass $m$ is projected with an initial velocity $u$ along a rough horizontal surface having a coefficient of kinetic friction $\mu$. In addition to friction, the block experiences an aerodynamic drag force $F_d = -kv$, where $k$ is a positive constant and $v$ is the instantaneous velocity of the block. Acceleration due to gravity is $g$. What is the total distance traveled by the block before it comes to rest?

A vehicle of mass $m = 1000 \text{ kg}$ is moving at $v_0 = 20 \text{ m/s}$ on a rough horizontal road, experiencing a constant braking friction force of $f = 8 \text{ kN}$. At $t=0$, it collides with a flexible barrier. The barrier exerts a constant retarding force of $F = 40 \text{ kN}$ on the vehicle for a duration of $\Delta t = 0.2 \text{ s}$, after which the vehicle loses contact with the barrier. What is the work done by the barrier on the vehicle during the collision?

A block of mass $m$ is projected with an initial velocity $v_0 = 4 \text{ m/s}$ on a rough horizontal surface. The coefficient of kinetic friction between the block and the surface varies with the distance $x$ from the starting point as $\mu = \mu_0(1 + \alpha x)$, where $\mu_0 = 0.1$ and $\alpha = 0.5 \text{ m}^{-1}$. Taking the acceleration due to gravity as $g = 10 \text{ m/s}^2$, what is the maximum distance $x_m$ the block travels before coming to rest?

A truck of initial mass $M_0 = 1000 \text{ kg}$ is moving with velocity $v_0 = 45 \text{ m/s}$ on a horizontal road. The coefficient of kinetic friction between the truck and the road is $\mu = 0.5$. The truck's cargo leaks vertically downwards at a constant rate $\lambda = 20 \text{ kg/s}$. If the deceleration process is governed by the generalized Newton's law $F_{\text{ext}} = \frac{dp}{dt}$ (where $p$ is the instantaneous momentum of the truck and its remaining cargo), calculate the total distance traveled by the truck before coming to rest. (Take $g = 10 \text{ m/s}^2$)

A particle of mass $m$ is released from rest from a height $h$ above a horizontal platform. Upon impact, the platform exerts an upward time-varying impulsive force $F(t) = k t^2$ on the particle, where $t$ is the time elapsed since the instant of first contact and $k$ is a positive constant. The particle remains in contact with the platform until the platform returns to its initial equilibrium position (i.e., the net displacement of the particle during the contact interval becomes zero). Assuming the effect of gravity during the extremely short contact time is negligible, what is the maximum height $H$ reached by the particle after it rebounds?

A rigid balloon of constant volume $V_0$ and mass $m = \rho_0 V_0$ is released from rest at a height $y = H$ in a static gas column. The gas column rests on the ground ($y=0$) where its pressure is $P_0$ and density is $\rho_0$. The gas obeys a polytropic relation $P = K \rho^2$ everywhere, where $K$ is a positive constant. Assuming the balloon's volume is small enough that the gas density is locally uniform around it, and ignoring viscous drag, what is the speed $v$ of the balloon just before it hits the ground? (Assume $H < \frac{2 P_0}{\rho_0 g}$)

Two fixed point charges, each of magnitude $+Q$, are located on the vertical $y$-axis at $y = +d$ and $y = -d$. A particle of mass $m$ and charge $+q$ is released from rest at $y = +d/2$. The region has a uniform downward gravitational field of acceleration $g$. Assuming no air resistance, what is the speed of the particle when it reaches the origin ($y = 0$)?

A spherical radioactive particle of constant radius $R$ and initial mass $m_0$ falls vertically from rest in a viscous fluid. The mass of the particle decreases according to the decay law $m(t) = m_0 e^{-\lambda t}$. Assume the decay process ejects the product particles such that they are left completely stationary in the lab frame immediately upon emission. The fluid exerts a drag force given by Stokes' Law, $F_d = 6\pi\eta R v$, where $\eta$ is the fluid viscosity and $v$ is the particle's velocity. Let $\beta = \frac{6\pi\eta R}{m_0}$. Which of the following differential equations correctly governs the velocity $v(t)$ of the particle?

A collimated light source, in its own rest frame, emits $N_0$ photons per second, each of frequency $\nu_0$. The source moves directly towards a stationary metal plate with a constant speed $v = \frac{3}{5}c$ (where $c$ is the speed of light). The metal plate has a work function $\phi = \frac{1}{2}h\nu_0$. Assuming all emitted photons strike the plate and the photoelectric quantum efficiency is $\eta$ (electrons emitted per incident photon), what is the stopping potential $V_s$ and the saturation photocurrent $I_s$ measured by an observer at rest with respect to the plate?

An incompressible, inviscid fluid of density $\rho$ flows steadily through a horizontal pipe of varying cross-section. The fluid velocity is $v_0$ at $x=0$ and $2v_0$ at $x=L$. A small particle of density $3\rho$ is released from rest at $x=0$. Neglecting viscous drag and added mass effects, and assuming the only horizontal force on the particle is due to the fluid's macroscopic pressure gradient, what is the velocity of the particle at $x=L$?

A block of mass $m$ is placed on the inclined surface of a rough wedge of angle $\theta = 37^\circ$. The wedge accelerates horizontally to the right with a time-varying acceleration $a(t) = 5t \text{ m/s}^2$ (where $t$ is in seconds). The wedge is oriented such that its inclined surface faces the right (i.e., the block slides down the incline towards the right). The coefficient of friction between the block and the wedge is $\mu = 0.5$. If the block is released from rest relative to the wedge at $t = 0$, at what time $t > 0$ does the block momentarily come to rest relative to the wedge? (Take $g = 10 \text{ m/s}^2$, $\sin 37^\circ = 3/5$, $\cos 37^\circ = 4/5$)

A parent nucleus $A$ of mass $4m$, initially at rest in the laboratory frame, undergoes a spontaneous decay into a nucleus $B$ of mass $3m$ and a particle $C$ of mass $m$, releasing a total kinetic energy $Q_1 = 6Q$. Subsequently, nucleus $B$ decays in flight into a nucleus $D$ of mass $2m$ and a particle $E$ of mass $m$, releasing a total kinetic energy $Q_2 = 3Q$. Assuming all motion is one-dimensional and non-relativistic, what is the maximum possible velocity of nucleus $D$ in the laboratory frame?

A small ball is dropped from a height $H = 20 \text{ m}$ above a large, horizontal platform initially at rest. The ball collides with the platform and rebounds with a coefficient of restitution $e = 0.5$. An automated sensor detects the impact and, after a reaction time delay $\Delta t = 0.5 \text{ s}$, causes the platform to accelerate vertically upwards with a constant acceleration $a = 4 \text{ m/s}^2$. Assuming the platform's movement does not interfere with the ball before $\Delta t$, what is the maximum height of the ball relative to the platform during its first rebound? (Take $g = 10 \text{ m/s}^2$ and neglect air resistance).

A uniform rigid door of mass $M = 6 \text{ kg}$ and width $L = 2 \text{ m}$ is hinged at one edge. At $t=0$, a sensor triggers an automated motor which applies a torque $\Gamma(t) = 12t^2 \text{ N}\cdot\text{m}$. Due to a reaction time gap of the release latch, the door remains locked and stationary until $t = 1 \text{ s}$. At exactly $t = 1 \text{ s}$, the latch instantaneously disengages. Assuming no friction in the hinge, what is the work done by the motor on the door during the interval $t=0$ to $t=3 \text{ s}$?

A block of mass $m = 4 \text{ kg}$ is initially at rest on a horizontal floor. At $t = 0$, a time-varying horizontal force $F(t) = 8t$ (where $t$ is in seconds and $F$ is in Newtons) is applied to the block. Due to a delayed engagement mechanism, the floor is perfectly frictionless for the first $0.5 \text{ s}$. At exactly $t = 0.5 \text{ s}$, the friction mechanism engages, and the floor instantly provides a coefficient of static friction $\mu_s = 0.4$ and kinetic friction $\mu_k = 0.2$. Taking $g = 10 \text{ m/s}^2$, what is the total distance traveled by the block from $t = 0$ to $t = 1.5 \text{ s}$?

A potentiometer wire of length $L = 10 \text{ m}$ has a uniform potential gradient $k = 2.0 \text{ V/m}$. An unknown cell of emf $E$ is connected in the secondary circuit with a galvanometer of resistance $R_g = 25 \ \Omega$ and inductance $L_g = 2.5 \text{ H}$. The jockey is moved along the wire from $x = 0$ at a constant velocity $v = 0.4 \text{ m/s}$. An automated sensor detects the instant the galvanometer current becomes zero and applies a brake to the jockey after a mechanical processing delay of $\Delta t_m = 0.15 \text{ s}$. The brake provides a constant deceleration $a = 0.8 \text{ m/s}^2$ until the jockey stops. If the final resting position of the jockey is $x_f = 4.50 \text{ m}$, what is the true emf $E$ of the cell? (Assume the jockey starts at $t=0$ from $x=0$ and the transient effects from the start of motion have decayed).

A stationary source at the origin $x=0$ begins emitting a continuous sound wave of frequency $f_0$ at $t=0$. The speed of sound in the medium is $c$. An observer, initially at $x = L$ at $t=0$, moves along the positive $x$-axis with a constant velocity $v$ ($v < c$). The observer's equipment is programmed to trigger a light flash exactly after detecting $N$ complete acoustic wave cycles from the source. What is the $x$-coordinate of the observer when the flash is triggered?

A block of mass $m = 1 \text{ kg}$ is attached to a light spring of force constant $k = 10 \text{ N/m}$ on a rough horizontal surface. The coefficients of static and kinetic friction are both $\mu = 0.2$. The block is pulled to a position $x_0 = 1.0 \text{ m}$ from the spring's natural length ($x=0$) and released from rest. Let $a_1$ be the magnitude of the block's acceleration immediately before it reaches its first turning point (instantaneous rest), and $a_2$ be the magnitude of its acceleration immediately after it reverses direction. What is the value of the ratio $\frac{a_1}{a_2}$? (Take $g = 10 \text{ m/s}^2$)

A simple pendulum of length $L$ and a bob of mass $m$ carrying charge $q$ is initially at rest in the vertical position ($\theta = 0$). A uniform horizontal electric field $E$ is suddenly switched on. The new equilibrium position of the pendulum makes a small angle $\theta_0$ with the vertical. The bob subsequently swings and reaches a maximum angular displacement $\theta_{max}$ from the vertical. Assuming small-angle approximations are valid, what is the angular velocity of the bob when its angular displacement from the vertical is $\theta = \frac{3}{4}\theta_{max}$?

A block of mass $M$ is projected vertically upward from the ground with an initial velocity $v_0$. The block is attached to one end of a uniform heavy flexible string of linear mass density $\lambda$. The string is initially coiled loosely on the ground. Assuming the string is sufficiently long and unfolds smoothly without tangling, what is the maximum height $h$ reached by the block? (Let $g$ be the acceleration due to gravity).

A particle of mass $m$ is initially at rest at $t=0$. It is subjected to a time-varying external driving force $F_{\text{ext}}(t) = m a_0 e^{-\frac{k}{m}t}$, where $a_0$ and $k$ are positive constants. The particle simultaneously experiences a drag force $F_d = -kv$, where $v$ is its instantaneous velocity. What is the magnitude of the maximum instantaneous power dissipated by the drag force during the motion?

A particle of mass $m = 1 \text{ kg}$ and charge $q = 1 \text{ C}$ is initially at rest at the origin. It is subjected to a time-varying electric field $\vec{E}(t) = E_x(t)\hat{i} + E_y(t)\hat{j}$. The components of the electric field are given by: $E_x(t) = \begin{cases} 6t & 0 \le t \le 1 \\ 3(3-t) & 1 < t \le 3 \\ 0 & t > 3 \end{cases}$ $E_y(t) = \begin{cases} 0 & 0 \le t \le 3 \\ 3(t-3) & 3 < t \le 5 \\ 6(6-t) & 5 < t \le 6 \\ 0 & t > 6 \end{cases}$ where $E_x$ and $E_y$ are in $\text{V/m}$ and $t$ is in seconds. What is the position vector of the particle at $t = 7 \text{ s}$?

A conducting rod of mass $m$ and length $l$ is placed on two long, parallel, frictionless horizontal conducting rails. The rails are connected at one end by an ideal inductor of inductance $L$. A uniform vertical magnetic field $B$ exists throughout the region. The resistance of the rod and the rails is negligible. At $t=0$, the rod is at rest. An external horizontal force $F(t)$ is applied to the rod such that the acceleration it would produce in the absence of the magnetic field is a constant $a_0$. What is the maximum energy stored in the inductor during the subsequent motion?

A particle undergoes damped one-dimensional oscillatory motion along the $x$-axis. Its acceleration $a$ is given by the differential equation $a = -\omega_0^2 x - b v$, where $\omega_0$ is the natural angular frequency, $b$ is a positive damping constant, $x$ is the displacement, and $v$ is the velocity. The particle is released from rest at $x = x_0$ ($x_0 > 0$) at $t = 0$ and first crosses the origin $x = 0$ at $t = t_1$. The algebraic area under the $a-t$ graph from $t = 0$ to $t = t_1$ is $-S_1$ ($S_1 > 0$), and the area under the $x-t$ graph for the same time interval is $S_2$. Which of the following correctly expresses the damping constant $b$?

A particle of mass $m = 2 \text{ kg}$ is moving along the positive x-axis. At $t = 0$, its velocity is $v_0 = 2 \text{ m/s}$. It experiences a net force in the direction of its motion given by $F = k v^2$, where $k = 0.5 \text{ kg/m}$ and $v$ is its instantaneous velocity. What is the average power delivered by this force to the particle during the time interval from $t = 0$ to $t = 1 \text{ s}$?

Block A of mass $m$ is attached to a linear spring of force constant $k$, with the other end of the spring fixed to a rigid wall. Block A rests on a smooth horizontal surface. Block B, also of mass $m$, is connected to Block A by a light, inextensible slack string of length $L$. Initially, the spring is relaxed and Block A is at rest. Block B is projected away from Block A with a horizontal velocity $v_0$ along the axis of the spring. At time $t = t_0$, the string becomes taut. What is the magnitude of the average tension in the string over the time interval from just before the string becomes taut ($t = t_0^-$) until the two-block system comes to instantaneous rest for the first time?

An ideal inductor of inductance $L$ (in Henry) and a resistor of resistance $R$ (in Ohm) are connected in series to a DC voltage source of constant EMF $E$ (in Volt) and a switch. The switch is closed at $t = 0 \text{ s}$. What is the average current $\langle i \rangle$ flowing through the circuit during the time interval from $t = 0$ to $t = \tau$, where $\tau = \frac{L}{R}$ is the inductive time constant of the circuit?

A non-linear circuit element $X$ has a voltage-current characteristic given by $V_X = \alpha I^2$, where $\alpha = 2 \ \text{V}\cdot\text{A}^{-2}$ and $I$ is the current flowing through it ($I > 0$). The element $X$ is connected in series with a resistor of resistance $R = 2 \ \Omega$, an ideal DC voltage source $V_{DC} = 12 \ \text{V}$, and an ideal AC voltage source $v_{ac}(t) = v_0 \sin(\omega t)$, where $v_0 = 60 \ \text{mV}$. Assuming $v_0$ is sufficiently small so that the small-signal linear approximation is valid, what is the amplitude of the AC component of the current flowing through the circuit?

Consider an ideal gas in thermal equilibrium at absolute temperature $T$. The gas consists of molecules, each of mass $m$. Let $k$ be the Boltzmann constant. Assuming the speeds $v$ of the molecules follow the standard Maxwell-Boltzmann speed distribution, which of the following expressions correctly represents the statistically averaged value of the inverse speed, $\langle v^{-1} \rangle$, of the gas molecules?

A particle of mass $m$ is initially at rest. From $t = 0$, it is driven in a straight line by an engine that delivers a constant power $P$. The particle experiences a resistive drag force $F_d = kv$, where $v$ is the instantaneous velocity and $k$ is a positive constant. Which of the following expressions correctly represents the velocity $v$ of the particle as a function of time $t$?

A block of mass $m$ is placed on a smooth horizontal surface and attached to a non-linear spring. The spring exerts a restoring force $F = -k_0 x |x|$, where $k_0$ is a positive constant (in $\text{N}\cdot\text{m}^{-2}$) and $x$ is the displacement of the block from its equilibrium position (in $\text{m}$). The block is displaced to a distance $x = A$ and released from rest. Which of the following expressions correctly gives the time period $T$ of the resulting oscillatory motion?

The velocity-time ($v-t$) graph of a particle moving along the x-axis exhibits changing concavity and is modeled by the equation $v(t) = 4t^3 - 12t^2 + 16$, where $v$ is in $\text{m/s}$ and $t$ is in seconds. Determine the average speed of the particle over the time interval from $t = 0$ to $t = 3 \text{ s}$.

The displacement-time ($x-t$) graph of a particle of mass $m = 2.0 \text{ kg}$ moving along the x-axis is described by the function $x(t) = \frac{x_0}{1 + (t/t_0)^2}$ for $t \ge 0$, where $x_0 = 4.0 \text{ m}$ and $t_0 = 3.0 \text{ s}$. The particle experiences a position-dependent conservative force $F(x)$. Let $x_c$ be the position where the $x-t$ graph changes its concavity. What is the effective spring constant $k_{\text{eff}} = -\frac{dF}{dx}$ evaluated at $x = x_c$?

Consider a crystalline solid where the interatomic potential energy $U(r)$ between two adjacent atoms separated by a distance $r$ is modeled as: $U(r) = U_0 + \frac{1}{2}c(r-r_0)^2 - g(r-r_0)^3$ where $r_0$ is the equilibrium separation at absolute zero temperature. The constants $c$ (in $\text{J}\cdot\text{m}^{-2}$) and $g$ (in $\text{J}\cdot\text{m}^{-3}$) are positive. The term $-g(r-r_0)^3$ represents the anharmonicity of the potential. Assume the anharmonic term is small compared to the thermal energy $kT$, where $k$ is the Boltzmann constant (in $\text{J}\cdot\text{K}^{-1}$) and $T$ is the absolute temperature (in $\text{K}$). Using classical statistical mechanics, determine the coefficient of linear thermal expansion $\alpha$ of the solid. (Hint: The average displacement $\langle x \rangle = \langle r - r_0 \rangle$ can be evaluated using the Boltzmann factor $e^{-U(x)/kT}$ and the approximation $e^{gx^3/kT} \approx 1 + \frac{gx^3}{kT}$. Standard integral: $\int_{-\infty}^{\infty} x^4 e^{-ax^2} dx = \frac{3}{4a^2}\sqrt{\frac{\pi}{a}}$)

An ideal monoatomic gas of molar mass $M$ (in $\text{kg mol}^{-1}$) is initially in a thermodynamic state with absolute temperature $T_0$ (in $\text{K}$) and volume $V_0$ (in $\text{m}^3$). The Maxwell-Boltzmann speed distribution function $f(v)$ for this gas exhibits two inflection points. Let $v_1$ be the speed corresponding to the higher inflection point. The gas is then compressed reversibly and adiabatically to a volume $\frac{V_0}{8}$. Let $v_2$ be the speed corresponding to the higher inflection point of the new distribution. Which of the following expressions correctly represents $v_2$? (Here $R$ is the universal gas constant in $\text{J K}^{-1} \text{mol}^{-1}$)

A series circuit consisting of an inductor of inductance $L$ (in Henrys) and a resistor of resistance $R$ (in Ohms) is initially open. At time $t = 0\text{ s}$, a switch is closed, connecting an alternating voltage source $V(t) = V_0 \cos(\omega t)$ (in Volts) across the circuit, where $\omega$ is the angular frequency (in rad/s). Assuming the initial current $I$ (in Amperes) in the inductor is zero, what is the value of the concavity of the current-time graph, defined as $\frac{d^2I}{dt^2}$, exactly at $t = 0^+$?