Category: Part 1: Mechanics

Problem Statement A stone is thrown vertically upward from a tower of height $H = 50\,\text{m}$ with speed $v_0 = 10\,\text{m/s}$. Simultaneously another stone is dropped from the top. Find when and where they meet. Given Information $H = 50\,\text{m}$ $v_0 = 10\,\text{m/s}$ Physical Concepts & Formulas This problem applies fundamental physics principles to the…

Problem Statement A boat’s speed in still water is $v$, river flows at $u = nv$ ($n=2$). At what angle to the stream should the boat head to minimize downstream drift? Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The…

Problem Statement Solve the kinematics problem: A body rotates from rest with $\beta = at$ ($a = 0.50\,\text{rad/s}^3$). When does the total acceleration vector of a body point make $\alpha = 60°$ with velocity? Angular velocity: $\omega = \int_0^t at’\,dt’ = \tfrac{at^2}{2}$ For a point at radius $R$: $w_\tau = \beta R = atR$, $w_n…

Problem Statement Solve the kinematics problem: A particle moves on the cardioid $r = a(1-\cos\varphi)$ with constant $\dot\varphi = \omega$. Find $v$ and $|\vec w|$ at $\varphi = 90°$. Polar velocity components: $v_r = \dot r$, $v_\varphi = r\dot\varphi$ $$\dot r = a\sin\varphi\cdot\dot\varphi = a\omega\sin\varphi$$ At $\varphi=90°$: $v_r = a\om Given Information See problem statement…

Problem Statement Solve the kinematics problem: A point moves on a circle of radius $R$ with constant angular acceleration $\beta = 0.10\,\text{rad/s}^2$ starting from $\omega_0 = 0.5\,\text{rad/s}$. Find the angle $\alpha$ between total acceleration and velocity at $t = 2.0\,\text{s}$. Angular velocity: $\omega = \omega_0+\beta t = 0.5+0.10(2.0) Given Information $\beta = 0.10\,\text{rad/s}$ $\omega_0 =…

Problem Statement Solve the kinematics problem: A wheel with $\beta = 3.0\,\text{rad/s}^2$ makes $n = 10$ revolutions in an interval during which $\omega$ doubles. Find the initial angular velocity. Let initial $\omega_1$, final $\omega_2 = 2\omega_1$. Total angle $\theta = 2\pi n = 20\pi$. Kinematic equation: $\omega_2^2 = \omega_1^2 + 2\beta\th Given Information $\beta =…

Problem Statement Solve the kinematics problem: A particle traces a cycloid: $x = a(\omega t – \sin\omega t)$, $y = a(1-\cos\omega t)$. Find (a) speed $v(t)$; (b) magnitude of acceleration. (a) Velocity: $$\dot x = a\omega(1-\cos\omega t),\quad \dot y = a\omega\sin\omega t$$ $$v = a\omega\sqrt{(1-\cos\omega t)^2+\sin^2\omega t} = a\omega\sqrt{2-2 Given Information See problem statement for…

Problem Statement A cannon fires a shell with initial velocity $v_0 = 240\,\text{m/s}$. Target is $l = 5.10\,\text{km}$ away at the same level. Find possible flight times (no air drag). Given Information $v_0 = 240\,\text{m/s}$ $l = 5.10\,\text{km}$ Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires…

Problem Statement Solve the kinematics problem: A particle starts at the origin with velocity field $\dot x = a$, $\dot y = bx$. Find the trajectory $y(x)$. Integrate: $x = at \implies t = x/a$ $$\dot y = bx = bat \implies y = \int_0^t bat’\,dt’ = \frac{bat^2}{2}$$ Substitute $t = x/a$: $$y = \frac{b}{2a}\cdot\frac{x^2}{1}…

Problem Statement Solve the kinematics problem: A body is thrown at angle $\alpha$ to the horizontal with initial speed $v_0$ (no air drag). Find: (a) flight time; (b) maximum height; (c) range; (d) speed at highest point. Initial components: $v_{0x}=v_0\cos\alpha$, $v_{0y}=v_0\sin\alpha$ (a) Flight time (return to $y=0$): $$T = \frac{2v_{0y}}{g} Given Information See problem statement…