Guided Solver
Choose an exercise and solve it step by step. The system checks every numerical answer.
Physics 1 23
Braking car
A car travels at v₀ = 72 km/h and brakes with constant deceleration a = −5 m/s².\nHow many metres does it travel before stopping?
Stone thrown from a bridge
A stone is thrown horizontally from a bridge h = 45 m high with initial velocity v₀ = 15 m/s.\nCalculate: (a) fall time, (b) horizontal range, (c) impact velocity.
Block on inclined plane with friction
A block of m = 5 kg is on an inclined plane at θ = 30°.\nThe kinetic friction coefficient is μ_d = 0.2. Calculate the acceleration of the sliding block.
Centripetal force — road curve
A car of m = 1200 kg takes a curve of radius R = 80 m at speed v = 60 km/h.\nCalculate the necessary centripetal force and the minimum static friction coefficient.
Ball on a slide — work-energy theorem
A ball of m = 0.2 kg starts from rest at the top of a slide h = 3 m high.\nNeglecting friction, calculate the speed at the bottom.
Elastic collision — two masses
Mass m₁ = 3 kg with velocity v₁ = 4 m/s collides elastically with m₂ = 1 kg at rest.\nCalculate the final velocities of both.
Archimedes' buoyancy — submerged object
An iron cube (ρ_iron = 7874 kg/m³) with side L = 0.1 m is completely submerged in water (ρ_H₂O = 1000 kg/m³).\nCalculate: (a) the volume, (b) the buoyant force, (c) whether it sinks or floats.
Bernoulli equation — tube with constriction
Water flows in a horizontal pipe. Section 1: A₁ = 0.04 m², v₁ = 2 m/s, P₁ = 1.5 × 10⁵ Pa.\nSection 2: A₂ = 0.01 m². Calculate v₂ and P₂.
Ideal gas — isobaric process
An ideal gas occupies V₁ = 2 L at T₁ = 300 K at constant pressure.\nWe heat it to T₂ = 450 K. Calculate the new volume V₂ and the work done.
Carnot cycle — efficiency
A Carnot engine operates between T_H = 500 K (hot source) and T_C = 300 K (cold source).\nIt absorbs Q_H = 1000 J per cycle. Calculate: (a) efficiency, (b) work produced, (c) heat rejected.
Spring and harmonic oscillator
A mass m = 0.5 kg is attached to a spring with k = 200 N/m.\nCalculate: (a) the natural angular frequency, (b) the period, (c) the frequency.
Simple pendulum — period
A simple pendulum has length L = 1 m. Calculate the period for small oscillations on Earth (g = 9.81 m/s²) and on the Moon (g_L = 1.62 m/s²).
Circular motion with variable acceleration
A point moves along a circle of radius R = 0.20 m, initially with angular velocity ω₀ = 0.5 rad/s. At t = 0 it accelerates with α = 0.15·t rad/s³. At t = 6 s find: angular velocity, total acceleration and the angle with the tangent.
Rotational kinetic energy of O₂
Find the average rotational kinetic energy of an oxygen molecule O₂ (diatomic) at T = 320 K. (k = 1.38×10⁻²³ J/K)
Calorimetry with melting ice
0.10 kg of ice at 0 °C is dropped into 0.50 kg of water at 25 °C (adiabatic container). Find the equilibrium temperature. (λ = 334000 J/kg, c = 4186 J/kg·K)
Ideal gas: temperature from PV = nRT
An ideal gas, n = 3 mol, occupies V = 8 L at pressure p = 8 atm. Find the temperature. (R = 8.314 J/mol·K, 1 atm = 101325 Pa)
Spring, rod and body (roto-translation)
A homogeneous rod and a point body, each of mass m₁ = m₂ = 2 kg, lie on a smooth horizontal plane. A spring (k = 40000 N/m) compressed by Δx = 5 cm sits between one end of the rod and the body. Find the speed v of the body after release.
Jumping the ditch (projectile motion)
A motorcyclist rides up a ramp inclined at α = 35° to jump a ditch d = 8 m wide. Find: minimum take-off speed, maximum height of the jump, and the angle of the velocity with the horizontal after 0.25 s.
Cannon recoil
A cannon of mass M = 600 kg, at rest, fires a projectile m = 3 kg at v_p = 120 m/s inclined at α = 25°. Find the recoil speed and the impulse of the ground's constraint reaction.
Disk rolling on an inclined plane
A homogeneous disk (M = 0.5 kg, R = 5 cm) rolls without slipping on a plane inclined at θ = 25° (μs = 0.4). Find: angular acceleration, friction force, and the maximum angle before slipping.
Hinged rod with a rope
A homogeneous rod AB (M = 8 kg, L = 1.2 m) is hinged at A and held horizontal by a rope at B making an angle α = 40° with the rod. Find: rope tension, vertical reaction at the pivot, and the angular velocity when — once the rope is cut — the rod passes through the vertical.
Simple pendulum
A simple pendulum of length L = 0.8 m is released from an angle θ₀ = 25°. Find the period (small oscillations) and the maximum speed at the lowest point.
Thermodynamic cycle (work and entropy)
n = 2 mol of an ideal gas at T₁ = 300 K perform a reversible cycle: isothermal compression from V₁ = 12 L to V₂ = 3 L, adiabatic expansion back to V₁, isochoric heating up to T₁. Find the work of the isotherm and the total entropy change over the cycle.
Physics 2 13
Coulomb force between two charges
Two point charges q₁ = +4 μC and q₂ = −6 μC are placed at distance r = 0.30 m in vacuum.\nCalculate: (a) the interaction force, (b) the electric field produced by q₁ at the location of q₂.
Charged sphere — field and potential
A conducting sphere of radius R = 5 cm carries total charge Q = 2 μC.\nCalculate: (a) the field E at r₁ = 10 cm from the outer surface, (b) the potential V on the surface, (c) the stored electrostatic energy.
Parallel-plate capacitor with dielectric
A parallel-plate capacitor has plates of area A = 400 cm² and separation d = 2 mm. It is filled with a dielectric of constant εᵣ = 5 and connected to V = 100 V.\nCalculate: (a) capacitance, (b) charge on the plates, (c) internal electric field E, (d) stored energy.
Magnetic field of a wire and solenoid
(a) An infinite straight wire carries current I = 8 A. Calculate the field B at distance r = 4 cm.\n(b) A solenoid with n = 1200 turns/m carries current I = 3 A. Calculate the internal field B.
Lorentz force — radius of trajectory
A proton (m = 1.673×10⁻²⁷ kg, q = 1.602×10⁻¹⁹ C) enters perpendicularly into a magnetic field B = 0.5 T with velocity v = 2×10⁶ m/s.\nCalculate: (a) the radius of the circular trajectory, (b) the revolution period, (c) the cyclotron frequency.
Solenoid — self-inductance and magnetic energy
A solenoid has N = 800 turns, length l = 40 cm, cross-section A = 12 cm² and carries current I = 5 A.\nCalculate: (a) self-inductance L, (b) magnetic energy U, (c) internal field B, (d) energy density u.
Induced EMF — rotating loop
A rectangular loop of area A = 200 cm² rotates with angular velocity ω = 120π rad/s in a magnetic field B = 0.3 T.\nCalculate: (a) the peak EMF, (b) the RMS value of the EMF, (c) the rotation frequency.
RLC circuit in alternating current
A series RLC circuit has R = 50 Ω, L = 0.2 H, C = 50 μF, powered at V = 220 V (RMS), f = 60 Hz.\nCalculate: (a) X_L and X_C, (b) total impedance Z, (c) RMS current I, (d) resonance frequency.
Electromagnetic wave — intensity and radiation pressure
A laser emits an EM wave with electric field amplitude E₀ = 500 V/m.\nCalculate: (a) average intensity, (b) amplitude B₀, (c) radiation pressure on an absorbing surface, (d) force on a mirror of area A = 1 cm² (total reflection).
Young's double-slit interference
In a Young's experiment with λ = 550 nm, the two slits are d = 0.40 mm apart, the screen is at L = 2.0 m.\nCalculate: (a) the fringe spacing, (b) the position of the 3rd maximum, (c) the position of the 2nd minimum.
Thin lens — image position and magnification
A converging lens has f = +15 cm. An object is placed at p = 25 cm from the lens.\nCalculate: (a) the image position q, (b) the transverse magnification m, (c) the type of image.
Photoelectric effect
UV light with λ = 180 nm strikes a caesium surface with φ = 2.0 eV.\nCalculate: (a) the photon energy in eV, (b) the maximum KE of the emitted electron, (c) the maximum speed of electrons, (d) the threshold frequency.
Wave-particle duality — de Broglie and Bohr
(a) An electron is accelerated by V = 1000 V. Calculate the de Broglie wavelength.\n(b) For hydrogen, calculate the radius of the 2nd Bohr orbit and the energy of level n=2.
Calculus 1 15
Domain of a composite function
Determine the domain of f(x) = ln(x² − 4) + √(9 − x²).
Inverse function — exponential and logarithm
Given f(x) = e^{3x−2}, find the inverse function f⁻¹(x) and its domain.
Notable limit with sin(ax)/(bx)
Calculate lim_{x→0} sin(4x)/(2x) using the fundamental notable limit.
Indeterminate form ∞/∞ — rational
Calculate lim_{x→+∞} (3x² − 2x + 1)/(x² + 5).
Continuity with parameter — sin(kx)/x
Find k such that f(x) = { sin(kx)/x for x≠0, 2 for x=0 } is continuous at x=0.
Intermediate value theorem — root of equation
Show that the equation x³ − 3x + 1 = 0 has at least one real root in the interval (1, 2) using Bolzano's theorem.
Derivative — product rule
Calculate the derivative of f(x) = x² · eˣ using the product rule.
Chain rule — composite function
Calculate the derivative of f(x) = sin(ln(x² + 1)) using the chain rule.
Indefinite integral — power and logarithm
Calculate the indefinite integral ∫ (3x² + 2/x) dx.
Definite integral — area under a curve
Calculate the area under f(x) = x² + 1 on the interval [0, 2].
Geometric series — convergence and sum
Calculate the sum of the geometric series Σ_{n=0}^{∞} (3/4)^n.
Root test — series convergence
Determine whether the series Σ_{n=1}^{∞} (n/(2n+1))^n converges using the root test.
Differential equation — separable variables
Solve the differential equation y' = 2x·y with initial condition y(0) = 3.
2nd order ODE — harmonic oscillator
Solve the differential equation y″ + 4y = 0 with initial conditions y(0) = 2, y′(0) = 0.
Power series: radius of convergence
Find the radius of convergence of .
Calculus 2 12
Geometric series — convergence and sum
Calculate the sum of the geometric series Σ_{n=0}^{∞} (3/4)^n.
Root test — series convergence
Determine whether the series Σ_{n=1}^{∞} (n/(2n+1))^n converges using the root test.
Separable variable ODE
Solve the ODE y' = 2xy with initial condition y(0) = 3.
Linear 1st order ODE
Solve y' + 3y = 6 with y(0) = 0.
2nd order ODE: characteristic equation
Find the general solution of y'' − 5y' + 6y = 0.
Double integral over a rectangle
Calculate on D = [0,1]×[0,2].
Gradient and tangent plane
For , calculate |∇f(3,4)| and write the tangent plane.
Lagrange — maximum of xy subject to x+y=4
Using Lagrange multipliers, maximise f(x,y)=xy subject to x+y=4.
Double integral in polar coordinates
Calculate on the disc x²+y²≤4.
Line integral — ∫_γ (x+y) ds
Calculate the line integral on γ(t)=(t,t), .
Divergence theorem
Calculate the flux of F=(x,y,z) through the sphere x²+y²+z²=R² with R=2.
Power series: radius of convergence
Find the radius of convergence of .
Chemistry 30
Moles and molar mass
Given 25.0 g of NaOH (M = 40.0 g/mol), calculate the number of moles and the mass corresponding to 0.500 mol.
Limiting reagent
For the reaction , you have 10.0 g of and 5.00 g of . Find the limiting reagent and the moles of produced.
M(N₂) = 28.0 g/mol, M(H₂) = 2.02 g/mol.
Percent yield
The reaction has a theoretical yield of 50.0 g of CaO. In the lab you obtain 42.5 g. Calculate the percent yield and the mass of CaCO₃ needed to obtain 50.0 g of CaO.
M(CaCO₃) = 100.1 g/mol, M(CaO) = 56.1 g/mol.
Bohr model — energy levels
Calculate the energy of level n=3 in hydrogen (En = -13.6/n² eV) and the n=3 → n=2 transition energy.
E2 = -3.40 eV (given).
Ionization energy
The ionization energy of hydrogen is 13.6 eV. Calculate the energy needed to ionize a hydrogen atom from level n=2.
En = -13.6/n² eV.
Quantum numbers and orbitals
Determine the number of orbitals in subshells and the electron capacity of energy levels.
Electronegativity difference and bond type
Determine the bond type (ionic, pure covalent, polar covalent) for Na-Cl (EN: Na=0.93, Cl=3.16), H-O (H=2.20, O=3.44), C-C (C=2.55).
Thresholds: ΔEN < 0.4 → pure cov.; 0.4–1.7 → polar cov.; > 1.7 → ionic.
Bond energy — bond length
Compare: C≡C (839 kJ/mol), C=C (614 kJ/mol), C-C (348 kJ/mol). Why is the triple bond shorter?
Lewis structures and formal charge
Draw the Lewis structure of nitrate ion NO₃⁻. Calculate the formal charge of each atom and determine the most stable resonance structure.
Valence: N=5, O=6, charge -1 = +1 e⁻.
Redox balancing — half-reaction method
Balance the redox reaction in acidic medium: MnO₄⁻ + Fe²⁺ → Mn²⁺ + Fe³⁺.
Standard potential and spontaneity
Determine if Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s) is spontaneous.
E°(Zn²⁺/Zn) = -0.76 V, E°(Cu²⁺/Cu) = +0.34 V.
Oxidation numbers
Determine the oxidation numbers of each element in: H₂SO₄, K₂Cr₂O₇, NaHCO₃.
Boyle's law — ideal gases
A gas occupies 2.50 L at 1.20 atm at constant temperature. Calculate the volume at 3.60 atm (Boyle: P₁V₁ = P₂V₂).
Ideal gas law
Calculate the temperature (in °C) of 0.500 mol of gas occupying 12.0 L at 1.80 atm.
R = 0.08206 L·atm/(mol·K). PV = nRT.
Charles's law — volume and temperature
A gas at 27 °C occupies 3.00 L. At constant pressure, calculate the volume at 127 °C (Charles: V₁/T₁ = V₂/T₂, T in Kelvin).
Reaction enthalpy calculation
Calculate ΔH° for: 2H₂(g) + O₂(g) → 2H₂O(l).
ΔH°f(H₂O(l)) = -285.8 kJ/mol.
Heat capacity and specific heat
How much heat is needed to heat 500 g of water from 20 °C to 80 °C?
cH₂O = 4.184 J/(g·°C). q = m·c·ΔT.
Hess's law
Calculate ΔH° for: C(s) + ½O₂(g) → CO(g) using:
(1) C(s) + O₂(g) → CO₂(g) ΔH° = -393.5 kJ
(2) CO(g) + ½O₂(g) → CO₂(g) ΔH° = -283.0 kJ
Latent heat of fusion
How much heat is needed to melt 250 g of ice at 0 °C?
ΔHfus = 334 J/g. q = m·ΔHfus.
Heating curve — ice to steam
Calculate total heat to bring 50.0 g of ice at -10 °C to steam at 110 °C.
cice = 2.09 J/(g·°C), ΔHfus = 334 J/g, cwater = 4.184 J/(g·°C), ΔHvap = 2260 J/g, csteam = 2.01 J/(g·°C).
Boiling point and pressure
Water boils at 100 °C at 1 atm. At 2000 m altitude (P ≈ 0.80 atm), ΔHvap = 40.7 kJ/mol. Use Clausius-Clapeyron: ln(P₂/P₁) = -(ΔHvap/R)(1/T₂ - 1/T₁). R = 8.314 J/(mol·K).
Estimate T₂.
Equilibrium constant K<sub>c</sub>
For: N₂(g) + 3H₂(g) ⇌ 2NH₃(g), equilibrium concentrations:
[N₂] = 0.50 M, [H₂] = 0.80 M, [NH₃] = 0.30 M. Calculate Kc.
Le Châtelier's principle
For N₂(g) + 3H₂(g) ⇌ 2NH₃(g) ΔH = -92 kJ.
Predict the effect of: (a) increasing [N₂], (b) increasing P, (c) increasing T.
pH of a weak acid
Calculate the pH of 0.100 M acetic acid (CH₃COOH, Ka = 1.8×10⁻⁵).
CH₃COOH ⇌ CH₃COO⁻ + H⁺.
Nernst equation
Calculate cell potential for Zn²⁺(0.010 M)/Zn: E° = -0.76 V.
Use Nernst: E = E° - (0.0592/n)·log(Q). T = 298 K.
Electrolysis — Faraday's law
How many grams of copper deposit at the cathode passing 2.50 A for 30.0 min through CuSO₄ solution?
Cu²⁺ + 2e⁻ → Cu(s). F = 96485 C/mol. M(Cu) = 63.55 g/mol.
Galvanic series — emf calculation
Calculate the standard emf of: Al(s) | Al³⁺(aq) || Cu²⁺(aq) | Cu(s).
E°(Al³⁺/Al) = -1.66 V, E°(Cu²⁺/Cu) = +0.34 V.
Determining molecular formula
A compound contains: C 54.52%, H 9.15%, O 36.33%. Molar mass is 132.16 g/mol. Determine empirical and molecular formulas.
Atomic masses: C=12.01, H=1.008, O=16.00.
Solubility and solubility product
Ksp of AgCl is 1.8×10⁻¹⁰. Calculate molar solubility of AgCl in water and in 0.010 M NaCl.
AgCl(s) ⇌ Ag⁺(aq) + Cl⁻(aq).
Stoichiometry — limiting reactant
React 10.0 g Zn with 20.0 mL of 6.00 M HCl.
Zn(s) + 2HCl(aq) → ZnCl₂(aq) + H₂(g).
M(Zn) = 65.38 g/mol. Determine limiting reactant and volume of H₂ produced (STP, 22.4 L/mol).
Chimica Organica 8
Degrees of unsaturation
Compute the degrees of unsaturation (DoU) of styrene C₈H₈.
Structural isomers of C₄H₁₀
How many structural isomers does the alkane C₄H₁₀ have?
Bond angle (sp hybridization)
In acetylene HC≡CH, what is the H—C—C bond angle? (sp carbon)
Number of stereoisomers
A molecule has 3 chiral centers and no internal symmetry. How many stereoisomers are possible?
Hückel rule
A planar, conjugated ring has 10 π electrons. For which value of n does it satisfy Hückel's rule (4n+2)?
Markovnikov rule
Addition of HBr to 2-methylpropene CH₂=C(CH₃)₂. On which carbon does the Br bond?
SN1 or SN2 mechanism?
A tertiary alkyl halide reacts with a weak nucleophile in a protic solvent. Which mechanism prevails?
Relative acidity
Which is more acidic: acetic acid (CH₃COOH) or trichloroacetic acid (CCl₃COOH)?