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Displacement is the straight-line vector from start to finish. The two legs are perpendicular, so the magnitude is the hypotenuse of a right triangle: the square root of (12 squared plus 5 squared), which equals the square root of 169, namely 13 m.
A scalar is fully described by magnitude alone, with no direction. Speed has only magnitude, whereas velocity, displacement and acceleration all require a stated direction to be complete.
With zero initial velocity, distance equals half the acceleration times the square of the time. Substituting gives 0.5 × 2 × 36 = 36 m. The velocity-time area is a triangle, which confirms this value.
For free fall from rest, speed equals g multiplied by the time. Using g = 10 m/s² and t = 3 s gives 30 m/s. Air resistance is neglected in this idealised case.
Newton's third law states that forces arise in pairs acting on different bodies. The action and reaction are always equal in magnitude and opposite in direction, regardless of the masses involved.
Power is the rate of doing work, so its unit is the joule per second, which is named the watt. One watt equals one joule of energy transferred each second.
Glass is optically denser than air, so light travels more slowly in it. By Snell's law the ray bends toward the normal when entering the denser medium, provided the incidence is oblique.
An ammeter is connected in series, so it must not add resistance to the circuit; an ideal one has zero resistance. A voltmeter is connected in parallel and must draw no current, so an ideal one has infinite resistance.
Newton's second law gives acceleration as net force divided by mass. Dividing 10 N by 4 kg yields 2.5 m/s². Because the floor is frictionless, the applied force is the net horizontal force.
At the highest point the velocity is zero, so the height equals the square of the initial speed divided by twice g. That gives 400 divided by 20, which is 20 m. Energy conservation gives the same result.
The horizontal component equals the force multiplied by the cosine of the angle to the horizontal. The cosine of 60° is 0.5, so the component is 10 N. The vertical component would be 17.3 N.
Total momentum is conserved in the collision. Taking the 2 kg body's direction as positive, the initial momentum is 2×4 − 3×2 = 2 kg·m/s. Dividing by the combined mass of 5 kg gives 0.4 m/s in the positive direction.
The work done equals weight times height, that is 500 N × 4 m = 2000 J. Dividing by the 2 s taken gives 1000 W. Constant speed means the lifting force equals the weight.
Centripetal force equals mass times the square of the speed divided by the radius. Substituting gives 2 × 36 ÷ 3 = 24 N. This force always points toward the centre of the circle.
For parallel resistors the reciprocals add: 1/6 + 1/3 = 1/2, so the equivalent resistance is 2 Ω. The result is always smaller than the smallest individual resistance.
Wave speed equals frequency multiplied by wavelength. Multiplying 50 Hz by 4 m gives 200 m/s. Frequency is the number of cycles per second and wavelength is the length of one cycle.
The thin-lens equation gives 1/v = 1/f − 1/u = 1/10 − 1/30 = 1/15, so v = 15 cm. A positive image distance for a converging lens means a real image on the side opposite the object.
Boyle's law states that at constant temperature the pressure of a fixed mass of gas is inversely proportional to its volume. Reducing the volume therefore raises the pressure, since the product of pressure and volume stays constant.
Heat equals mass times specific heat capacity times temperature rise. Multiplying 2 kg by 4200 J/(kg·K) by 10 K gives 84 000 J. The specific heat capacity is the energy per kilogram per kelvin.
Electric force equals charge multiplied by field strength. Converting 5 μC to 5 × 10⁻⁶ C and multiplying by 200 N/C gives 1 × 10⁻³ N. The force direction follows the field for a positive charge.
With air resistance neglected, the only force is the weight, which acts vertically. There is therefore no horizontal acceleration, so the horizontal velocity component remains constant while the vertical component changes uniformly.
The period of a simple pendulum is proportional to the square root of length divided by g. Quartering g multiplies the period by the square root of 4, which is 2, so the new period is 2T. The mass of the bob does not affect the period.
Stretching at constant volume doubles the length and halves the cross-sectional area. Resistance is proportional to length and inversely proportional to area, so the two effects compound to give four times the original resistance.
Einstein's photoelectric equation gives the stopping potential as (hf − φ)/e. This is linear in frequency with slope h/e and intercept −φ/e, so the slope is Planck's constant divided by the electronic charge. The slope is the same for all metals.
The number of half-lives is 24 divided by 8, which is 3. Each half-life halves the remaining nuclei, so the surviving fraction is one half cubed, equal to one eighth.
A scalar is fully described by magnitude alone, with no direction needed. Work is the product of force and displacement along the force direction and is a scalar quantity. Displacement, velocity and acceleration are all vectors because each requires a direction for complete description.
For uniform acceleration from rest, distance equals half the acceleration multiplied by the square of the time: s = ½at². Substituting a = 2 m/s² and t = 5 s gives s = ½ × 2 × 25 = 25 m. The final speed would be 10 m/s, but distance depends on the average speed of 5 m/s over 5 s.
Weight varies inversely with the square of the distance from the Earth's centre. At a height equal to one Earth radius, the distance from the centre doubles, so weight falls to one quarter of 50 N, that is 12.5 N. The loss in weight is therefore 50 − 12.5 = 37.5 N.
Momentum is conserved, so the common speed is (2 × 6)/(2 + 4) = 2 m/s. Initial kinetic energy is ½ × 2 × 36 = 36 J and final kinetic energy is ½ × 6 × 4 = 12 J. The loss is therefore 36 − 12 = 24 J, which is not among the listed values, so the correct listed figure is the one matching this computation: 12 J is the final kinetic energy, and the loss is 24 J.
For rolling without slipping, acceleration down an incline is g sin θ divided by (1 + I/MR²). The solid sphere has I = (2/5)MR², giving the smaller denominator and thus the greater acceleration. The hollow cylinder has the larger moment of inertia factor, so it rolls more slowly and arrives later.
Elastic potential energy in a spring is ½kx². Substituting k = 200 N/m and x = 0.1 m gives ½ × 200 × 0.01 = 1 J. Note that the energy depends on the square of the extension, not linearly on it.
For an incompressible fluid in steady flow, the equation of continuity states that area times speed is constant. If the area falls to one third, the speed must rise by a factor of three to keep the volume flow rate fixed. Pressure in the narrow section correspondingly drops, as described by Bernoulli's principle.
Capillary rise is given by h = 2T cos θ/(ρgr). For water, ρ = 1000 kg/m³, so r = 2 × 0.07/(1000 × 10 × 0.05) = 0.14/500 = 2.8 × 10⁻⁴ m. This shows that a smaller tube radius produces a larger rise.
Work done by a gas is pressure times the change in volume. At constant volume the change in volume is zero, so no work is done by or on the gas. Pressure, temperature and internal energy all rise as heat is supplied at constant volume.
For an ideal gas at constant temperature, the change in internal energy is zero. The first law then requires that heat absorbed equals the work done by the gas. This is why an isothermal expansion requires a continuous supply of heat.
In a standing wave, nodes occur where the displacement is always zero and are separated by half a wavelength. Antinodes lie midway between nodes, also half a wavelength apart. The full wavelength spans the distance between three consecutive nodes.
When a source approaches a stationary observer, successive wavefronts are emitted from positions closer to the observer. This shortens the observed wavelength, so the observer hears a higher frequency. The effect is the Doppler shift, familiar as the higher pitch of an approaching vehicle.
The critical angle satisfies sin C = 1/n for a glass-air boundary. With C = 30°, sin 30° = 0.5, so n = 1/0.5 = 2.0. A higher refractive index produces a smaller critical angle.
For thin lenses in contact, the powers add: P = 1/f₁ + 1/f₂. Here P = 1/0.10 − 1/0.20 = 10 − 5 = 5 dioptres, giving f = 1/5 = 0.20 m, that is +20 cm. The combination remains converging because the positive lens is stronger.
In a uniform field the forces on the two equal and opposite charges of a dipole are equal in magnitude and opposite in direction. They therefore cancel, giving zero net force. The dipole does, however, experience a torque that tends to align it with the field.
For capacitors in series, the reciprocals add: 1/C = 1/3 + 1/3 + 1/3 = 1 μF⁻¹, so C = 1 μF. The equivalent capacitance in series is always smaller than the smallest individual capacitor.
The magnetic force on a current-carrying wire is F = BIL sin θ. With θ = 90°, force per unit length is BI = 0.5 × 2 = 1.0 N/m. The direction is given by Fleming's left-hand rule.
For a sinusoidal supply, the rms value is the peak value divided by the square root of two. Thus 310/1.414 is approximately 220 V. This is why mains supplies quoted as 220 V have peak values near 310 V.
The magnetic force on a moving charge is qvB sin θ, where θ is the angle between velocity and field. When the velocity is parallel to the field, sin θ is zero, so no magnetic force acts. The particle therefore continues in a straight line at constant speed.
For an ideal transformer, the voltage ratio equals the turns ratio: Vs/Vp = Ns/Np. Thus Vs = 220 × 200/1000 = 44 V. A step-down transformer reduces voltage while increasing the available current.
Einstein's photoelectric equation gives maximum kinetic energy as photon energy minus work function. Here 3.5 − 2.0 = 1.5 eV. If the photon energy were below the work function, no emission would occur at all.
The Bohr radius of the nth orbit is proportional to n², so the first orbit is smallest and higher orbits are much larger. The energy of the nth level, by contrast, is proportional to −1/n². This explains why the energy levels crowd together at high n.
An alpha particle is a helium nucleus with mass number 4 and atomic number 2. Emitting it reduces the mass number by 4 and the atomic number by 2. Starting from mass number 240, the daughter has mass number 236 and atomic number 90, so among the listed pairs the mass number must be 236 and the atomic number 2 less than the parent.
The strong nuclear force is attractive at typical nuclear separations but becomes strongly repulsive when nucleons are brought very close together. This repulsive core prevents nuclei from collapsing. The force is short-ranged and acts between all nucleons, not only protons.
Total energy is ½kA², and potential energy at displacement x is ½kx². Setting potential energy equal to half the total energy gives x² = A²/2, so x = A/√2, about 0.707A. At this point the speed is 1/√2 of its maximum value.