Motion in a Plane - Class 11 Part 1

A steady current density $\vec{J} = J_0 \frac{x}{R} \hat{j}$ is maintained within a spherical region of radius $R$ centered at the origin, where $J_0$ is a positive constant. What is the magnetic field $\vec{B}$ at the origin?

A rigid semi-circular wire of radius $R$ is kept in the $xy$-plane. At time $t=t_0$, its ends are located at $(R, 0)$ and $(-R, 0)$ with the wire forming the curve $y = \sqrt{R^2 - x^2}$. The wire is moving with a constant velocity $\vec{v} = v_0 \hat{i}$. A magnetic field $\vec{B} = \alpha t x \hat{k}$ exists in the region, where $\alpha$ is a positive constant. What is the magnitude of the motional EMF induced across the wire at $t = t_0$?

A long taut string is aligned along the negative $x$-axis ($x < 0$). At $x=0$, the string is attached to a massless ring that is constrained to slide without friction along a rigid, straight rod. The rod lies in the $y$-$z$ plane and makes an angle $\theta$ with the positive $y$-axis. An incident transverse wave, polarized purely along the $y$-axis, is given by $\vec{D}_i(x,t) = A \sin(kx - \omega t) \hat{j}$. Which of the following represents the reflected wave $\vec{D}_r(x,t)$?

A block of mass $M = 2m$ is free to slide on a smooth horizontal floor. A uniform rod of mass $m$ and length $L$ is smoothly hinged at its lower end to the block. Initially, the system is at rest with the rod hanging vertically downwards. A constant horizontal force $F$ ($F > mg$) is applied to the free top end of the rod. Find the angular velocity $\omega$ of the rod at the instant it reaches the horizontal position.

A rigid conducting rod $OP$ of length $L$ rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about the origin $O$ in the $xy$-plane. The region contains a static, non-uniform magnetic field given by: $\vec{B} = B_0 \left[ \sin\left(\frac{\pi x}{L}\right) \cosh\left(\frac{\pi y}{L}\right) \hat{i} - \cos\left(\frac{\pi x}{L}\right) \sinh\left(\frac{\pi y}{L}\right) \hat{j} + \left(\frac{x^2 + y^2}{L^2}\right) \hat{k} \right]$ An observer $O'$ moves with a constant velocity $\vec{v}_0 = \omega L \hat{i} + \omega L \hat{j}$ relative to the laboratory frame. What is the magnitude of the motional electromotive force (EMF) across the rod $OP$ as measured by the observer $O'$?

Two particles, $A$ and $B$, are projected simultaneously in a uniform gravitational field $\vec{g} = -g \hat{j}$. Particle $A$ is projected from the origin $(0,0)$ with initial velocity $\vec{u}_A = u_0 \hat{i} + u_0 \hat{j}$. Particle $B$ is projected from the point $(d, 0)$ with initial velocity $\vec{u}_B = 2u_0 \hat{j}$. Both particles experience an air drag force $\vec{F} = -m \gamma \vec{v}$, where $m$ is the mass of the particle, $\vec{v}$ is its velocity, and $\gamma$ is a positive constant. If $d = \frac{3u_0}{\gamma}$, what is the distance of closest approach between the two particles?

A long horizontal plank is moving on a smooth horizontal floor with an initial velocity $v_0 \hat{i}$ and a constant acceleration $-a_0 \hat{i}$ ($a_0 > 0$). A solid cylinder of mass $m$ and radius $R$ is gently placed on the plank at $t=0$ with zero initial velocity and zero angular velocity. The coefficient of kinetic friction between the cylinder and the plank is $\mu$. Let $P$ be the top-most point of the cylinder and $Q$ be the right-most point of the cylinder (at the same height as the center of mass). Determine the magnitude of the velocity of point $P$ relative to point $Q$ at the exact instant the cylinder begins pure rolling on the plank.

An elevator is accelerating with an acceleration $\vec{a} = 5\sqrt{3} \hat{i} + 5 \hat{j} \text{ m/s}^2$, where $\hat{i}$ is along the horizontal floor of the elevator and $\hat{j}$ is vertically upwards. Inside this elevator, a projectile is fired from the origin on the floor with an initial speed of $20 \text{ m/s}$ at an angle of $60^\circ$ with the horizontal floor. Taking $g = 10 \text{ m/s}^2$ in the $-\hat{j}$ direction, what is the horizontal range of the projectile on the floor of the elevator?

An atom of mass $m$ is injected into a gas medium at $t=0$ with an initial velocity $v_0 \hat{i}$ relative to the lab frame. The gas medium flows with a uniform steady velocity $u_0 \hat{j}$ relative to the lab frame. The gas exerts a viscous drag force $\vec{F} = -b \vec{v}_{\text{rel}}$ on the atom, where $b$ is a positive constant and $\vec{v}_{\text{rel}}$ is the velocity of the atom relative to the gas. Exactly at $t=0$, the atom emits a photon of energy $E$ in the $+x$ direction. Assuming the atom does not emit any further photons, what is the total displacement vector of the atom relative to the gas medium from $t=0$ to $t \to \infty$? (Here $c$ is the speed of light in vacuum).

A river flows uniformly with velocity $v$ in the $+x$ direction. A wave generator produces plane surface waves that travel with a phase speed $c = 2v$ relative to the water. The waves travel downstream and reflect normally off a stationary vertical pier at $x=0$. The superposition of the incident and reflected waves creates an interference pattern consisting of a stationary spatial envelope and a propagating carrier wave. A swimmer swims downstream with a velocity $u = \frac{v}{2}$ relative to the river. What is the velocity of the swimmer relative to the phase velocity of the carrier wave?

A vessel is in the shape of a paraboloid of revolution, such that its cross-sectional area at a height $y$ from its bottom is given by $A(y) = k y$, where $k$ is a positive constant. The vessel is completely filled with an ideal, non-viscous liquid up to a height $H$. It is placed on a horizontal table of height $h$ from the ground. A small hole of cross-sectional area $a$ ($a \ll kH$) is opened at the bottom of the vessel, and the liquid shoots out horizontally. What is the time-averaged horizontal range of the liquid on the ground until the vessel completely empties?

A particle of mass $m$ and positive charge $q$ is projected from the origin $(0,0)$ in the $xy$-plane with an initial speed $u = \sqrt{2gL}$ at an angle $\theta = 45^\circ$ to the positive $x$-axis. The $y$-axis points vertically upwards. In addition to uniform gravity $g$ acting downwards, there exists an electric field $\vec{E} = E_0 \left( \frac{x}{L} \hat{i} - \frac{y}{L} \hat{j} \right)$ in the region. If the charge-to-mass ratio of the particle is $\frac{q}{m} = \frac{g}{E_0}$, what is the $x$-coordinate of the particle when it first returns to the $x$-axis ($y=0$)?

A particle of mass $m$ is projected from the ground with an initial speed $u$ at an angle $\theta$ to the horizontal. It experiences a velocity-dependent drag force $\vec{F}_d = -m \beta \vec{v}$, where $\beta$ is a positive constant and $\vec{v}$ is the instantaneous velocity. If the terminal velocity of the particle is $V_T$ and $u \ll V_T$, which of the following expressions best approximates the horizontal range of the projectile? (Assume acceleration due to gravity is $g$ and neglect terms of order $(u/V_T)^2$ and higher).

A projectile of mass $3M$ is fired from the origin on level ground with an initial speed $u = 50 \text{ m/s}$ at an angle $\theta = \sin^{-1}(4/5)$ above the horizontal. At the highest point of its trajectory, an internal explosion breaks it into two fragments of masses $M$ and $2M$. Immediately after the explosion, the lighter fragment $M$ moves such that it lands exactly at the point of projection. Assuming the ground is horizontal and $g = 10 \text{ m/s}^2$, at what distance from the point of projection does the heavier fragment land?

A train carriage is accelerating horizontally on a straight track with a constant acceleration $a = \frac{g}{\sqrt{3}}$, where $g$ is the acceleration due to gravity. A passenger inside the carriage projects a ball from the horizontal floor with an initial speed $u$ relative to the carriage, at an angle $\alpha$ with the floor, in the direction of the carriage's acceleration. If the ball has the same horizontal range on the floor for two distinct angles of projection $\alpha_1$ and $\alpha_2$, find the sum $\alpha_1 + \alpha_2$.

A particle of mass $m$ and charge $q > 0$ is projected from the origin in the $x-y$ plane with an initial speed $u$. The $y$-axis points vertically upwards. A uniform electric field $\vec{E}$ exists in the region such that the equipotential surfaces are vertical planes parallel to the $y-z$ plane, and the electric potential decreases in the positive $x$-direction. It is given that the magnitude of the electric force on the particle is exactly equal to its weight ($q|\vec{E}| = mg$). The particle is projected at an angle $\theta$ to the positive $x$-axis. If $\theta$ is chosen such that the particle lands on the $x$-axis at the maximum possible distance from the origin, what is the ratio of the particle's kinetic energy just before impact to its initial kinetic energy?

A rigid, massless plank is hinged to a horizontal, non-expanding floor. The plank is supported by a vertical cylindrical pillar of height $H$ at a horizontal distance $L$ from the hinge, forming an incline of angle $\alpha = 30^\circ$. The pillar is made of a material with coefficient of linear thermal expansion $\beta$ and Poisson's ratio $\nu$. It is tightly fitted inside a rigid, unexpandable cylindrical sleeve that prevents any lateral expansion. A small block of mass $m$ is launched from the hinge up the incline by a spring of stiffness $k$ compressed by $x_0$. The block travels as a projectile in the vertical plane, making an angle $\theta = 30^\circ$ with the incline. If the temperature of the pillar is increased by a small amount $\Delta T$, find the fractional change in the range of the projectile along the incline, $\frac{\Delta R}{R}$. Assume the spring and floor are unaffected by the temperature change.

An infinitely long straight wire carrying a uniform positive linear charge density $\lambda$ is fixed along the horizontal bottom edge of a smooth non-conducting inclined plane of inclination $\theta = 30^\circ$. A particle of mass $m$ and positive charge $q$ is projected from a point on the incline at a distance $r_0$ from the wire. The initial velocity of the particle is $v_0$, directed at an angle $\alpha = 60^\circ$ to the wire, along the inclined plane. If the system parameters satisfy the relation $\frac{q\lambda}{2\pi\epsilon_0} = m g r_0$ (where $g$ is the acceleration due to gravity), what is the speed of the particle when its distance from the wire becomes $2r_0$?

Two identical particles $A$ and $B$, each of mass $m$, are connected by a light inextensible string of length $L$. The system is falling freely under gravity. At a certain instant, the string is horizontal and the velocity of both particles is $v_0$ vertically downwards. At this exact instant, particle $A$ strikes a fixed smooth inclined plane of inclination $\theta$ to the horizontal. The collision of $A$ with the incline is perfectly inelastic in the direction normal to the plane. Mass $B$ is positioned such that the horizontal velocity acquired by $A$ is directed away from $B$, causing the string to immediately jerk. Determine the loss in total kinetic energy of the system immediately after the impact.

A large closed tank contains a liquid of density $\rho$. The liquid level is at a height $H$ above a small hole in the side of the tank. The gas above the liquid is maintained at a constant gauge pressure $P_0$. A spherical droplet of radius $r$ and density $\rho$ emerges horizontally from the hole into the surrounding air. The droplet experiences a viscous drag force $\vec{F}_d = -6\pi\eta r \vec{v}$, where $\eta$ is the viscosity of air and $\vec{v}$ is the droplet's velocity vector. Neglecting the buoyant force from the air, what is the horizontal displacement $x$ of the droplet at the exact instant its downward vertical velocity reaches $v_y$? (Assume acceleration due to gravity is $g$).

A hollow cylinder of mass $M$ contains a smooth, airtight piston of mass $m$. The cylinder is closed at its upper end and open at its lower end. Initially, the piston is locked, confining an ideal gas of volume $V_0$ and pressure $P_0$ between the closed upper end and the piston. The entire system is projected up a smooth inclined plane of inclination $\theta$ with an initial velocity $v_0$. Immediately after launch, the piston is unlocked. The gas expands, undergoing a polytropic process $PV^n = \text{constant}$ ($n > 1$). At the instant the center of mass of the system has traveled a distance $D$ up the incline, the volume of the gas is $rV_0$. Assuming the piston has not yet left the cylinder, which of the following expressions gives the velocity of the cylinder up the incline at this instant?

A small bead of mass $m$ (in $\text{kg}$) and charge $q > 0$ (in $\text{C}$) is threaded on a long, straight, frictionless horizontal wire lying along the $x$-axis. The entire system is immersed in a region containing a time-varying electric field $\vec{E} = E_0 \hat{i} + \alpha t^3 \hat{j}$ (in $\text{V}\cdot\text{m}^{-1}$) and a non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$ (in $\text{T}$), where $E_0$ and $B_0$ are positive constants, $\alpha$ is an unknown constant, and $t$ is time (in $\text{s}$). The bead is released from rest at the origin at $t=0$. If the normal force exerted by the wire on the bead is zero at all times, what is the value of the constant $\alpha$?

A point particle of mass $m$ is projected from a radial distance $r = R$ with an initial speed $v_0$. The initial velocity vector makes an angle of $\frac{\pi}{4}$ with the inward radial vector. The particle moves under the influence of a central attractive potential $U(r) = -\frac{m R^2 v_0^2}{4 r^2}$, which arises from a non-uniform field density. Assuming conservation of angular momentum, find the radial distance $r$ at which the velocity vector makes an angle of $\frac{\pi}{3}$ with the inward radial vector.

A smooth massive plate is moving with a constant velocity $\vec{V} = (3 \hat{i} + 4 \hat{j}) \text{ m/s}$. The plate's surface is a horizontal plane with its normal vector pointing along the positive y-axis. A small particle of mass $m = 2 \text{ kg}$ strikes the plate with a velocity $\vec{u} = (5 \hat{i} - 6 \hat{j}) \text{ m/s}$. If the coefficient of restitution for the collision is $e = 0.5$, find the work done by the impulsive normal force on the particle during the collision.

A particle of mass $m$ is moving on a smooth horizontal table, attached to a string passing through a small hole in the table. Initially, the particle moves in a circle of radius $R$ with speed $v_0$. At $t=0$, the string is pulled downwards such that the radial distance of the particle from the hole decreases at a constant rate $u = \frac{2v_0}{\pi}$. What is the magnitude of the change in the velocity vector of the particle from the instant just after the pulling begins until its radial distance becomes $R/2$?

A particle of mass $m$ and charge $q > 0$ is projected from the origin $(0,0,0)$ with an initial velocity $\vec{v} = v_0 \hat{i}$. The region $x > 0$ contains a non-uniform magnetic field given by $\vec{B} = \alpha x \hat{k}$, where $\alpha$ is a positive constant. The region $x < 0$ has no magnetic field. Neglecting gravity, what is the maximum positive x-coordinate ($x_{max}$) reached by the particle?

A wedge of angle $\theta = 45^\circ$ is forced to accelerate horizontally to the right with a time-varying acceleration $a(t) = kt$ ($k > 0$). A small block of mass $m$ rests on its rough inclined surface. The coefficients of static and kinetic friction between the block and the wedge are $\mu_s = 0.5$ and $\mu_k = 0.25$, respectively. Assuming the wedge's acceleration remains continuous, what is the magnitude of the sudden change in the vertical acceleration of the block at the exact instant it begins to slip? (Let $g$ be the acceleration due to gravity)

An ideal gas consisting of molecules of mass $m$ and number density $n$ is confined in a cylinder by a piston. The piston moves outward with a constant, small velocity $u$ ($u \ll \bar{v}$, where $\bar{v}$ is the mean speed of the gas molecules). Assuming the gas remains in thermal equilibrium at temperature $T$ and the collisions of the molecules with the piston are perfectly elastic, which of the following expressions gives the pressure $P$ exerted by the gas on the piston? (Let $P_0 = n k_B T$ be the static pressure of the gas).

A transverse sinusoidal wave travels along a taut string of linear mass density $\mu = 0.1 \text{ kg/m}$ in the positive $x$-direction. The wave propagation speed is $v = 10 \text{ m/s}$. The maximum transverse acceleration of any particle on the string is $a_{\text{max}} = 200 \text{ m/s}^2$, and the amplitude of the wave is $A = 2 \text{ cm}$. What is the time-averaged power $\langle P \rangle$ transmitted by the wave?

A particle of mass $m$ and charge $q > 0$ is moving in a circular orbit of radius $r_0$ with a constant speed $v_0$ in the $xy$-plane. For $t < 0$, the region contains a uniform and constant magnetic field $\vec{B} = B_0 \hat{k}$. At $t = 0$, the magnetic field begins to increase linearly with time as $\vec{B}(t) = B_0(1 + \alpha t) \hat{k}$, where $\alpha > 0$ is a small constant. Find the magnitude of the radial component of the jerk vector (defined as the rate of change of the acceleration vector, $\vec{j} = \frac{d\vec{a}}{dt}$) of the particle exactly at $t = 0^+$. Assume non-relativistic motion.

A geostationary satellite consists of a main body of mass $2m$ and a payload of mass $m$ connected by a massless rigid arm of length $L$. The center of mass of the system is in a stable circular orbit of radius $R$ around the Earth with an orbital angular velocity $\omega$. An internal motor causes the arm to rotate in the orbital plane such that the payload revolves around the main body with a constant angular speed $2\omega$ relative to the inertial frame. Which of the following represents the maximum magnitude of the absolute acceleration of the main body?

A hollow rigid sphere of inner radius $R$ is completely filled with an incompressible liquid of density $\rho$. The sphere rotates with a constant angular velocity $\omega$ about a vertical diameter. A small hole at the highest point of the sphere keeps the liquid pressure there equal to the atmospheric pressure $P_0$. Assuming the acceleration due to gravity is $g$, what is the net upward force exerted by the liquid on the upper half of the sphere?

A small block of mass $m$ is placed on a rough horizontal turntable rotating with a constant angular velocity $\omega$ about a vertical axis. At $t=0$, the block is at a distance $R$ from the axis and is given an initial velocity $v_0$ directed radially outward relative to the turntable. If the coefficient of kinetic friction between the block and the turntable is $\mu_k$, what is the radius of curvature of the block's trajectory in the reference frame of the turntable at $t=0$?

A small block of mass $m$ is projected with an initial velocity $v_0$ along the inside of a fixed, rough vertical circular track of radius $R$ from its lowest point. The coefficient of kinetic friction between the block and the track varies with the angle $\theta$ (measured from the lowest point) as $\mu(\theta) = \mu_0 \sin\theta$, where $\mu_0$ is a positive constant. Find the minimum value of $v_0^2$ required for the block to just complete the vertical circle.

An elevator accelerates vertically upwards with a constant acceleration $a = g$. Inside the elevator, a fixed vertical circular track of radius $R$ is completely submerged in a stationary, incompressible ideal fluid of density $\sigma$. A small spherical shell of negligible mass and volume $V$, completely filled with another incompressible fluid of density $\rho = 3\sigma$, is projected tangentially from the lowest point of the track's interior. Assuming no viscous drag, what is the minimum initial speed $v_0$ relative to the track required for the sphere to just complete the vertical loop?

A non-uniform rod of mass $M$ and length $L$ is pivoted freely at its upper end $O$. The linear mass density of the rod varies as $\lambda(x) = cx$, where $x$ is the distance from the pivot $O$ and $c$ is a positive constant. The rod is initially hanging vertically at rest. A horizontal impulsive force $J$ is applied at the bottom end of the rod. What is the minimum value of $J$ required so that the rod completes a full vertical circle about the pivot $O$?

A particle of mass $m$ is attached to a light inextensible string of length $R$ and suspended from a fixed point. It is projected horizontally from the lowest point with an initial speed $v_0$. During its motion in the vertical plane, it experiences a non-conservative tangential drag force $F_d = - \frac{k m}{2R} v^2$, where $v$ is its instantaneous speed and $k$ is a positive dimensionless constant. By recognizing the isomorphism of its kinetic energy evolution to a driven dissipative circuit, determine the minimum initial speed $v_0$ required for the particle to just complete the full vertical circle.

A uniform cylindrical roll of tape of initial mass $M_0$ and initial radius $R_0$ is mounted on a fixed, frictionless horizontal axle. The free end of the tape is pulled with a constant horizontal force $F$. As the tape unwinds, it does not slip or stretch, and its mass per unit length is $\lambda$. The roll can be modeled as a uniform solid cylinder at all times. Assuming the unwound portion of the tape moves horizontally without sagging, which of the following expressions gives the angular velocity $\omega$ of the roll when its radius has decreased to $r$?

A smooth straight horizontal tube is fixed on a horizontal turntable rotating with a constant angular velocity $\omega$ about a vertical axis passing through a point $O$. The shortest distance from $O$ to the tube is $d$. Let $C$ be the point on the tube closest to $O$. A particle of mass $m$ is released from rest relative to the tube at a point $A$ inside the tube, where the distance $AC = d$. Find the work done by the tube on the particle in the ground frame by the time its distance from $O$ becomes $2d$.

A rigid physical pendulum consists of a uniform circular disc of mass $M$ and radius $R$, pivoted freely at a point $P$ on its circumference. A point mass of mass $m = \frac{M}{2}$ is firmly attached to the disc at a point diametrically opposite to the pivot $P$. The pendulum is held such that the diameter passing through the pivot and the point mass is horizontal, and then released from rest. What is the magnitude of the linear velocity of the point mass when the pendulum reaches its lowest vertical position?

A particle of mass $m$ and positive charge $q$ is projected from the origin at $t=0$ with an initial speed $u$ at an angle $\theta$ to the horizontal. The particle moves in a region where a uniform downward gravitational field $g$ and a time-varying vertical electric field $\vec{E}(t) = E_0 \sin(\omega t) \hat{j}$ are present. The $y$-axis is taken vertically upwards. Given that $qE_0 < mg$, which of the following transcendental equations correctly determines the time of flight $T$ of the particle?

A small projectile of initial mass $m_0$ and constant charge $+q$ is launched from the origin over a horizontal ground with initial speed $v_0$ at an angle $\theta$ to the horizontal. A uniform electric field $E$ exists in the $+x$ (horizontal) direction, and uniform gravity $g$ acts in the $-y$ (vertical) direction. During its flight, the projectile ejects uncharged dust with zero relative velocity, such that its remaining mass at time $t$ is given by $m(t) = m_0 e^{-\lambda t}$. Which of the following expressions gives the horizontal range of the projectile? (Let $T = \frac{2v_0 \sin\theta}{g}$ be the time of flight)

A particle of mass $m$ is projected vertically upwards from the ground with an initial velocity $u = v_t$, where $v_t = \sqrt{mg/k}$ is the terminal velocity of the particle in the medium. The medium exerts a drag force $F_d = -k v^2$, where $v$ is the instantaneous velocity and $k$ is a positive constant. What is the total time of flight of the particle before it returns to the ground?

A cylindrical space station of radius $R$ rotates with constant angular velocity $\omega$ about its longitudinal axis in deep space. Two identical particles, A and B, each of mass $m$, are initially at rest on the inner surface. At $t=0$, particle A is projected radially inwards with a speed $u$ relative to the station, while particle B remains attached to the surface. Both particles act as sources of matter waves. The phase accumulated by a particle is given by $\phi = S/\hbar$, where $S = \int L dt$ is the classical action evaluated in the inertial frame. What is the magnitude of the phase difference $|\Delta \phi| = |S_A - S_B|/\hbar$ between the two particles at the instant particle A strikes the inner surface again?

A thin uniform ring of mass $m$ and radius $R$ is released from rest at a vertical height $h = 30 \text{ m}$ on a rough downward incline. It rolls without slipping until it reaches the lowest point of the track (height $0$). At the lowest point, the track smoothly transitions via a short frictionless circular fillet to a frictionless upward incline of angle $\theta = 30^\circ$ with the horizontal. The upward incline ends abruptly at a vertical height $H = 10 \text{ m}$ above the lowest point, from where the ring becomes a projectile in the air. Neglecting air resistance, find the total time of flight of the ring from the moment it leaves the track until it hits the horizontal ground (which is at height $0$). Take $g = 10 \text{ m/s}^2$.

A solid planet of radius $R$ has a spherically symmetric density distribution given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)$, where $r$ is the distance from the center of the planet and $\rho_0$ is a positive constant. A small particle is projected radially outward from the center of the planet through a narrow, smooth tunnel. If $g_s$ is the acceleration due to gravity at the surface of the planet, what is the minimum initial speed $v_0$ required for the particle to just reach the surface?

A small ball is projected from the origin $(0,0)$ over level ground with an initial velocity $\vec{u} = 30\hat{i} + 40\hat{j} \text{ m/s}$. At the highest point of its trajectory, it strikes the smooth inclined face of a massive wedge. The wedge is moving with a constant velocity $\vec{v}_w = 10\hat{j} \text{ m/s}$. The outward normal to the wedge's inclined face is $\hat{n} = \frac{-\hat{i} + \hat{j}}{\sqrt{2}}$. If the coefficient of restitution for the collision is $e = 0.5$, what is the maximum height from the ground reached by the ball? (Take $g = 10 \text{ m/s}^2$)

A time-varying periodic current $i(t)$ flows through a fixed resistor of resistance $R$. The current waveform has a period of $2t_0$ and is defined as: $i(t) = \begin{cases} I_0 \left(\frac{t}{t_0}\right)^{3/2} & \text{for } 0 \le t < t_0 \\ 0 & \text{for } t_0 \le t < 2t_0 \end{cases}$ where $I_0$ is a positive constant in Amperes, $t_0$ is in seconds, and $R$ is in Ohms. What is the average thermal power dissipated in the resistor over a long time?

A small bead of mass $m$ and positive charge $q$ is constrained to move with a constant speed $v_0$ along a semicircular wire of radius $R$ in the $xy$-plane. The wire connects the points $(R, 0, 0)$ and $(-R, 0, 0)$ passing through $(0, R, 0)$. The region contains a non-uniform magnetic field $\vec{B} = B_0 y \hat{k}$, where $B_0$ is a positive constant. What is the time-averaged magnetic force exerted on the bead as it travels from $(R, 0, 0)$ to $(-R, 0, 0)$?

A radioactive sample initially contains $N_0$ identical nuclei at $t=0$, each with a decay constant $\lambda$. Let $N_D(t)$ be the random variable representing the exact number of nuclei that have decayed by time $t$. Which of the following expressions correctly represents the statistical variance of $N_D(t)$?