Category: Part 1: Mechanics

Problem Statement Solve the kinematics problem: A body is launched at $v_0$, angle $\alpha$. At what time $t$ does the velocity vector make angle $\beta$ below the horizontal? Find the distance from launch at that instant. Velocity components: $v_x = v_0\cos\alpha$, $v_y = v_0\sin\alpha – gt$ Condition for angle $\beta$ below horizontal: $$\tan\b Given Information…

Problem Statement Three particles at the corners of an equilateral triangle of side $a$ simultaneously start chasing each other (each toward the next) at constant speed $v$. Find: (a) time to meet; (b) distance each travels. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles…

Problem Statement A point moves along the curve $y = a\sin(px)$ at constant speed $v$. Find the acceleration and radius of curvature at the crests. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation…

Problem Statement Solve the kinematics problem: Two particles move on a circle of radius $a$: positions $(a\cos\omega t,\, a\sin\omega t)$ and $(a\cos(\omega t+\phi),\, a\sin(\omega t+\phi))$. Find separation $l$ and relative speed $\tilde v$. Separation: $$\Delta x = a[\cos(\omega t+\phi)-\cos\omega t] = -2a\sin\!\left(\omega t+\tfrac\phi2\right Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Solve the kinematics problem: A point moves according to $x = at$, $y = bt^2$ ($a,b>0$). Find the normal and tangential accelerations and the radius of curvature at $t = 0$. Velocity: $v_x = a$, $v_y = 2bt$, $v = \sqrt{a^2+4b^2t^2}$ Acceleration: $w_x=0$, $w_y=2b$, $|\vec w| = 2b$ (constant) At $t=0$: $v=a$, $\vec…

Problem Statement Solve the kinematics problem: Two particles of equal mass move along the same line with velocities $v_1=1.0\,\text{m/s}$ and $v_2=2.0\,\text{m/s}$. Find the velocity of the centre of inertia $\tilde v$, and the velocity of each particle relative to it. Centre of inertia velocity (equal masses $m$): $$\tilde v = \frac{mv_1+mv_2}{ Given Information $v_1=1.0\,\text{m/s}$ $v_2=2.0\,\text{m/s}$…

Problem Statement Solve the kinematics problem: A point on a circular arc of radius $R$ starts from rest with constant tangential acceleration $w_\tau = b$. Find the total acceleration as a function of time $t$. Speed at time $t$: $v = bt$ Normal acceleration: $w_n = v^2/R = b^2t^2/R$ Total acceleration: $$|\vec{w}| = \sqrt{w_\tau^2+w_n^2} =…

Problem Statement A point moves in plane $xy$: $x = a\sin\omega t$, $y = a(1-\cos\omega t)$. Find: (a) distance traversed in time $\tau$; (b) angle between velocity and acceleration. Given Information See problem statement for all given quantities. Physical Concepts & Formulas Circular motion requires a centripetal force directed toward the centre, providing the centripetal…

Problem Statement Solve the kinematics problem: A point moves along a circle of radius $R$. Its speed varies with distance as $v = a\sqrt{s}$ ($a$ = const). Find the angle $\alpha$ between total acceleration and velocity. Tangential acceleration (rate of speed change): $$w_\tau = \frac{dv}{dt} = \frac{dv}{ds}\cdot v = \frac{a}{2\sqrt{s}}\cdot a\s Given Information See problem…

Problem Statement Solve the kinematics problem: A motorboat going downstream overcame a raft at point $A$; $\tau = 60\,\text{min}$ later it turned back and after some time passed the raft at a distance $l = 6.0\,\text{km}$ from point $A$. Find the flow velocity of the river. Key idea: Work in the reference frame of the…