Category: Part 1: Mechanics

Problem Statement Solve the kinematics problem: A point moves on a circle of radius $R = 20\,\text{cm}$ with constant speed, making one revolution in $T = 0.40\,\text{s}$. Find: (a) the modulus of mean velocity over $t = T/4$; (b) the modulus of mean acceleration over $t = T/4$. Linear speed: $v = 2\pi R/T =…

Problem Statement Solve the Newton’s Laws / mechanics problem: A body is thrown from the foot of an incline of angle $\beta$ with initial velocity $v_0$. At what angle $\alpha$ (measured from the incline) gives maximum range along the slope? Set up axes along and perpendicular to incline. Effective gravity components: $$g_\parallel = g\sin\beta,\quad g_\perp…

Problem Statement A body falls from height $h$ with air drag $F_{drag} = \alpha v$. It reaches speed $v_0$ just before impact. Find $h$ (qualitative approach since terminal velocity is relevant). 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 is thrown at $\theta_0 = 45°$, $v_0 = 25\,\text{m/s}$. Find the radius of curvature at (a) the highest point; (b) the point of launch. (a) At the apex: Speed: $v = v_0\cos45° = 25/\sqrt2\,\text{m/s}$ Centripetal acceleration $= g$ (only gravity, perpendicular to horizontal velocity): $$R_{ape Given Information…

Problem Statement A particle decelerates with $w = -k\sqrt{v}$ from $v_0$. Find the stopping distance $s$ and stopping time $T$. 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 laws and equations of motion,…

Problem Statement Solve the kinematics problem: A body starts from rest with $w = at$ ($a = 0.50\,\text{m/s}^3$). Find: (a) mean velocity over first $t_0 = 4.0\,\text{s}$; (b) instantaneous velocity at $t_0/2$. Velocity: $v(t) = \int_0^t at’\,dt’ = \tfrac12 at^2$ Position: $x(t) = \int_0^t \tfrac12 at’^2\,dt’ = \tfrac{a}{6}t^3$ (a) Mean velocity: Given Information $a =…

Problem Statement Solve the kinematics problem: A particle decelerates with $w = -\alpha v^2$ ($\alpha>0$) starting at $x=0$, $v=v_0$. Find: (a) $v(x)$; (b) $x(t)$. (a) $v\,dv/dx = -\alpha v^2 \implies dv/dx = -\alpha v$ $$\int_{v_0}^v \frac{dv’}{v’} = -\alpha x \implies \ln(v/v_0) = -\alpha x$$ $$\boxed{v = v_0 e^{-\alpha x}}$$ (b) $dx/dt = v_0e Given Information…

Problem Statement Solve the kinematics problem: Particle 1 moves along $x$-axis with constant acceleration $a$ from rest. Particle 2 moves along $y$-axis with constant velocity $b$. Find the angle $\theta$ that the velocity of particle 2 relative to particle 1 makes with the $x$-axis at time $t$. Velocities: $\vec v_1 = at\hat x$, $\vec v_2…

Problem Statement Solve the kinematics problem: A particle starts at $x=0$ with speed $v_0$. Acceleration $w = -\alpha x$. Find $v(x)$. Use $w = v\,dv/dx$: $$v\frac{dv}{dx} = -\alpha x$$ $$\int_{v_0}^v v’\,dv’ = -\alpha\int_0^x x’\,dx’$$ $$\frac{v^2-v_0^2}{2} = -\frac{\alpha x^2}{2}$$ $$\boxed{v = \sqrt{v_0^2 – \alpha x^2}}$$ This is SHM with $\o Given Information See problem statement for…

Problem Statement Solve the kinematics problem: A man walking at $v_1 = 3\,\text{m/s}$ in rain that falls vertically at $v_2 = 6\,\text{m/s}$. At what angle to the vertical does the rain appear to fall relative to the man? How fast does the apparent rain move? Rain velocity in ground frame: $\vec v_{rain} = -v_2\hat j…