Choose an exercise and solve it step by step. The system checks every numerical answer.
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?
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.
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.
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.
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.
Mass m₁ = 3 kg with velocity v₁ = 4 m/s collides elastically with m₂ = 1 kg at rest.\nCalculate the final velocities of both.
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.
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.
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.
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.
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.
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).
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.
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.
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.
(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.
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.
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.
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.
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.
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).
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.
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.
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.
(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.
Determine the domain of f(x) = ln(x² − 4) + √(9 − x²).
Given f(x) = e^{3x−2}, find the inverse function f⁻¹(x) and its domain.
Calculate lim_{x→0} sin(4x)/(2x) using the fundamental notable limit.
Calculate lim_{x→+∞} (3x² − 2x + 1)/(x² + 5).
Find k such that f(x) = { sin(kx)/x for x≠0, 2 for x=0 } is continuous at x=0.
Show that the equation x³ − 3x + 1 = 0 has at least one real root in the interval (1, 2) using Bolzano's theorem.
Calculate the derivative of f(x) = x² · eˣ using the product rule.
Calculate the derivative of f(x) = sin(ln(x² + 1)) using the chain rule.
Calculate the indefinite integral ∫ (3x² + 2/x) dx.
Calculate the area under f(x) = x² + 1 on the interval [0, 2].
Calculate the sum of the geometric series Σ_{n=0}^{∞} (3/4)^n.
Determine whether the series Σ_{n=1}^{∞} (n/(2n+1))^n converges using the root test.
Solve the differential equation y' = 2x·y with initial condition y(0) = 3.
Solve the differential equation y″ + 4y = 0 with initial conditions y(0) = 2, y′(0) = 0.
Find the radius of convergence of ∑n=1∞n xn3n\displaystyle\sum_{n=1}^\infty \dfrac{n\,x^n}{3^n}n=1∑∞3nnxn.
Solve the ODE y' = 2xy with initial condition y(0) = 3.
Solve y' + 3y = 6 with y(0) = 0.
Find the general solution of y'' − 5y' + 6y = 0.
Calculate ∬D(x+y) dA\displaystyle\iint_D (x+y)\,dA∬D(x+y)dA on D = [0,1]×[0,2].
For f(x,y)=x2+y2f(x,y)=x^2+y^2f(x,y)=x2+y2, calculate |∇f(3,4)| and write the tangent plane.
Using Lagrange multipliers, maximise f(x,y)=xy subject to x+y=4.
Calculate ∬D(x2+y2) dA\displaystyle\iint_D (x^2+y^2)\,dA∬D(x2+y2)dA on the disc x²+y²≤4.
Calculate the line integral ∫γ(x+y) ds\int_\gamma(x+y)\,ds∫γ(x+y)ds on γ(t)=(t,t), t∈[0,1]t\in[0,1]t∈[0,1].
Calculate the flux of F=(x,y,z) through the sphere x²+y²+z²=R² with R=2.
Given 25.0 g of NaOH (M = 40.0 g/mol), calculate the number of moles and the mass corresponding to 0.500 mol.
For the reaction N2+3H2→2NH3\mathrm{N_2 + 3H_2 \to 2NH_3}N2+3H2→2NH3, you have 10.0 g of N2\mathrm{N_2}N2 and 5.00 g of H2\mathrm{H_2}H2. Find the limiting reagent and the moles of NH3\mathrm{NH_3}NH3 produced.M(N₂) = 28.0 g/mol, M(H₂) = 2.02 g/mol.
The reaction CaCO3→CaO+CO2\mathrm{CaCO_3 \to CaO + CO_2}CaCO3→CaO+CO2 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.
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).
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.
Determine the number of orbitals in subshells and the electron capacity of energy levels.
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.
Compare: C≡C (839 kJ/mol), C=C (614 kJ/mol), C-C (348 kJ/mol). Why is the triple bond shorter?
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⁻.
Balance the redox reaction in acidic medium: MnO₄⁻ + Fe²⁺ → Mn²⁺ + Fe³⁺.
Determine if Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s) is spontaneous.E°(Zn²⁺/Zn) = -0.76 V, E°(Cu²⁺/Cu) = +0.34 V.
Determine the oxidation numbers of each element in: H₂SO₄, K₂Cr₂O₇, NaHCO₃.
A gas occupies 2.50 L at 1.20 atm at constant temperature. Calculate the volume at 3.60 atm (Boyle: P₁V₁ = P₂V₂).
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.
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).
Calculate ΔH° for: 2H₂(g) + O₂(g) → 2H₂O(l).ΔH°f(H₂O(l)) = -285.8 kJ/mol.
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.
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
How much heat is needed to melt 250 g of ice at 0 °C?ΔHfus = 334 J/g. q = m·ΔHfus.
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).
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.
For: N₂(g) + 3H₂(g) ⇌ 2NH₃(g), equilibrium concentrations:[N₂] = 0.50 M, [H₂] = 0.80 M, [NH₃] = 0.30 M. Calculate Kc.
For N₂(g) + 3H₂(g) ⇌ 2NH₃(g) ΔH = -92 kJ.Predict the effect of: (a) increasing [N₂], (b) increasing P, (c) increasing T.
Calculate the pH of 0.100 M acetic acid (CH₃COOH, Ka = 1.8×10⁻⁵).CH₃COOH ⇌ CH₃COO⁻ + H⁺.
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.
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.
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.
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.
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).
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).
Uningenia