IB Physics Topic A Tutorial Sheet: Harder Projectile Motion

A tutorial sheet of harder projectile motion questions.

1. A horizontal tunnel has a vertical height of 3.0 m. A ball is thrown inside the tunnel with an initial speed of 18.0 m s^-1. What is the greatest horizontal distance that the ball can travel before it bounces for the first time?
2. A ball is thrown at 20.0 m s^-1 up an inclined plane of angle of elevation 30.0°. Find the maximum range of the ball on the inclined plane.
3. The ball in question 2 is now thrown down the inclined plane. Explain why the range on the inclined plane is now greater.

IB Physics Topic A Tutorial Sheet : Block on an Incline

A tutorial sheet on Topic A SL/HL that practices the concepts of motion, momentum, force and energy. There are “show that” questions that require a labelled diagram with all steps shown and “explain” questions that require elaboration involving the laws of Physics to obtain full marks in each case.

A block of mass m is placed on the hypotenuse of a smooth wedge of mass M and base angle 𝜃 that rests on a smooth horizontal surface.

The wedge is held at rest by a horizontal force F acting perpendicular to its vertical side.

1. Show that the acceleration of the block of mass m as it slides down the wedge is gsin𝜃.
2. Show that the normal force between the sliding mass and the wedge is mgcos𝜃.
3. Show that the normal force between the horizontal surface and the wedge is g(M+m cos²𝜃)
4. Show that F = mgcos𝜃 sin𝜃.
5. Show that the time taken by the block to slide a distance L from rest is t = √( 2L/( g sin𝜃 ) ).

The block stays at rest relative to the wedge when a new force F acts perpendicular to the vertical side of the wedge.

1. Show that the acceleration of the block is the same as the acceleration of the wedge.
2. Show that the normal force between the block and the wedge is N = mg/cos𝜃.
3. Show that the resultant force acting on the block is Nsin𝜃.
4. Show that the acceleration of the block is gtan𝜃.
5. Show that F = (M+m)gtan𝜃.

The block is released from rest on the smooth stationary wedge which has a hypotenuse of length L.

1. Is linear momentum conserved as the block slides down the wedge? Explain why the wedge moves to the left if the block has a velocity component to the right.
2. Explain why kinetic energy is not conserved as the block slides down the wedge.
3. Explain why energy is conserved as the block slides down the wedge.
4. The block is released from rest at a vertical height H. At a certain instant the height of the block is h and the horizontal and vertical velocity components of the block and wedge are (Vx,Vy) and (W,0) respectively. Write down the equations for conservation of linear momentum and energy.
5. HL Show that the acceleration of the wedge is - mg sin𝜃 cos𝜃/(M + m sin²𝜃 ).
6. HL Show that the time taken for the block to travel down the wedge is √2L ( (M + m sin²𝜃) /(g (M+m) sin𝜃) ). Show that the limiting case of m/M → 0 gives t = √( 2L/( g sin𝜃 ) ).

IB Physics Static Friction

A tutorial sheet on static friction, a concept in the HL and SL courses that is often misunderstood. These are “show that” questions requiring all steps with explanation and a labelled diagram to obtain full marks.

A rough cube of edge length a and weight W is placed on a rough horizontal surface. An identical second rough cube of weight W is placed on top of the first cube. The coefficient of static friction between each surface is µ. A horizontal force F is applied perpendicular to the cubes at a height h above the top of the bottom cube. Show that

1. the lower cube does not slide along the horizontal plane if F < 2µW,
2. the upper cube does not slide over the lower cube if F < µW,
3. the lower cube does not topple if F < aW/(a+h),
4. the upper cube does not topple if F < aW/(2h).

With h = a/2 the force F is now increased steadily from zero. Show that

1. If µ < 2/3 equilibrium is disturbed by the upper cube sliding,
2. If µ > 2/3 equilibrium is disturbed by the lower cube toppling.

IB Physics Centrifugal force?

A tutorial sheet on “centrifugal force”. Is this a “real” force? Some usages of this term are taken from some well known books. What do the authors mean?

1. Space Frontier , 1971 by Wernher von Braun, page 49. A circular orbit occurs whenever a small mass travelling through the gravitational field of a big one, happens to have a speed at which the centrifugal force is precisely strong enough to balance the large body's gravitational pull.

IB HL Physics Spacetime Diagrams

A tutorial sheet on spacetime diagrams follows.

1. Define the term worldline.
2. Describe the path of a photon on a spacetime diagram.
3. Are unit intervals on each spacetime axis in the same inertial reference frame the same length when measured with a ruler?
4. Are unit intervals on each spacetime axis in the different inertial reference frames the same length when measured with a ruler?
5. A particle P moves at a constant velocity 0.8c relative to an inertial reference frame S. Draw the worldline of P in the reference frame of S.
6. A particle Q moves at a constant velocity 0.8c relative to an inertial reference frame S'. Reference frame S' is moving at a constant velocity of 0.6c relative to a reference frame S. Draw the worldline of Q in the reference frame S.

IB Physics Negative signs in Physics

In many equations in Physics there are negative signs. Sometimes answers have a negative sign. What does this mean? A tutorial sheet on these concepts follows.

1. A particle moving in a straight line has a positive velocity and a negative acceleration. In words, describe the movement of the particle and sketch graphs showing its displacement, velocity and acceleration in terms of time.
2. In the equation for heat conduction there is a negative sign. What does this mean?
3. HL In Lenz's law there is a negative sign. What does this mean?
4. HL extension. In the time dependent Schrodinger equation there is the imaginary unit. What does this mean?
5. HL extension. The equation of continuity in fluid flow is div 𝜌_mv = -𝜌_m,_t . State the meaning of these symbols and the negative sign.
6. HL extension. The equation of continuity in charge flow is div J = -𝜌_c,_t . State the meaning of these symbols and the negative sign.

IB Physics Conservation of linear momentum

A tutorial sheet on linear momentum.

1. State the law of conservation of linear momentum.
2. In the collision of two particles the change in momentum of one particle is 2 Ns south-east. Find the change in momentum of the other particle.
3. A ball of mass m falls from rest at a height h and bounces vertically to a height h/2. Find the momentum gained by the Earth in the collision.

IB Physics why does light bend when it enters water?

A tutorial sheet on refraction.

1. State the definition of refraction.
2. Explain why light bends towards the normal when it enters water from air.
3. Explain why sound bends away from the normal when it travels from air to water.

IB Physics Work Done in Moving a Charge

A tutorial sheet on the HL topic of fields.

1. A charge +Q is fixed at rest. Find the work done in moving a charge -q from a distance d1 to a distance d2 along the line joining the charges.
2. Two charges -q are at rest an infinite distance apart. Find the work done in moving each charge inwards along the line joining the charges so they are finally a distance d apart.

IB Physics Heat and Internal Energy

A tutorial sheet on the words heat and internal energy, concepts that are often confused in examination answers.

1. The internal energy of an object is the sum of the potential energies of molecular interaction and the kinetic energies of molecular movement.
2. The internal energy of an object can be changed by doing work on it or by supplying heat to it.
3. Heat is energy that is transmitted between two objects at a different temperature.
4. By doing work and supplying heat the change in internal energy is given by the first law of thermodynamics. ΔU = Δq + W.
5. When a paddle wheel does work in churning water the energy supplied did not become heat but became internal energy of the water due to its rise in temperature. This energy may be transmitted to the surroundings as heat energy and will become internal energy of the surroundings.

IB HL Physics Tetrahedron Problems

A sheet of questions where the geometry of the problem involves a regular tetrahedron.

1. Six resistors R are placed as the sides of a regular tetrahedron. Show that the total resistance across two corners of the tetrahedron is R/2.
2. Four point charges are placed at the vertices of a regular tetrahedron. Show that the magnitude of the resultant force acting on one of the charges is √6 x the force between two of the charges.
3. A solid uniform tetrahedron of mass M and side L is rotated about an axis through a vertex that is perpendicular to the opposite side. Show that the moment of inertia of the tetrahedron about this axis is M L²/20.
4. Four masses M are placed at the vertices of a regular tetrahedron of side L and released from rest. Show that the time taken for the masses to meet is π √( L³/(32G M) ).
5. A triangular pyramid is a regular tetrahedron of mass M and side L. Find the work done in lifting the material from ground level to build the triangular pyramid.
6. A regular tetrahedron of side L and density 𝜌₁ floats vertically with its base downwards in a fluid of density 𝜌₂, 𝜌₁< 𝜌₂. Find the period of small vertical oscillations of the tetrahedron when 𝜌₁/ 𝜌₂ = 2/5 , g = 9.8 m s^-2 and L = 100 m.

IB Physics Does the Mass of an Atom Change when it emits Light?

A tutorial sheet that explores the relationship between mass and energy when an atom emits light. Some explanatory notes are given from the article by Ralph Baierlein, “Teaching E=mc2, An Exploration of Some Issues” published in The Physics Teacher, March 1991, 170-173.

1. When an atom emits a photon the atom's energy decreases.
2. When the energy of the atom decreased so did its inertia (mass).
3. Changes in mass and energy occur in parallel. There is no conversion of one into the other.
4. As Einstein stated, if the energy changes by ΔE then the mass changes in the same sense by Δm/c².
5. An atom of mass m is at rest. The atom releases a photon of frequency f. Show that the mass of the atom after it releases the photon is m √( 1 - 2 hf/mc² ). Explanation.Four momentum is conserved. The first component of the four momentum vector is energy. The second component of the four momentum vector is linear momentum. mc²= hf + √(p²c²+m₁²c²) and hf/c=p. Solve simultaneously to find m₁ by eliminating p.
6. The electrons, protons and neutrons in the atom do not change mass. The released energy comes from the electron in the atom making a transition to a lower energy state.

IB Physics Average Speed Problems

A tutorial sheet on average speed problems.

1. A car travels at 50 km h^-1 for a distance of 150 km and then travels at 30 km h^-1 for another 150 km. What is the average speed for the entire trip?
2. A car travels to a town at an average speed of 40 km h^-1. The return journey on the same road is made with an average speed of 30 km h^-1. Find the average speed for the entire journey.
3. Two identical planes Mike and Ike race against each other. They start at the same place and both have an air speed of v km h^-1. A wind from the east of w km h^-1 blows, w < v. The planes start at the same point and time. Mike travels to a point d km east on the ground and returns. Ike travels to a point d km north on the ground and returns. Show that the average speed of Mike is (v² - w²)/v and the average speed of Ike is (v² - w²)^1/2.
4. Show that Ike always wins the race. Hint (v² - w²)^1/2 < v, v² - w² < v(v² - w²)^1/2. This problem is taken from Relativity for the Layman by James A Coleman, 1958, pp 32-33 and the solution is given in The Physics Teacher, vol 34, Oct 1996, pp 438-439.

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IB Physics Entropy Change of Two Objects in Contact

A tutorial sheet on entropy change. Entropy change questions are often asked in HL Physics.

1. If a quantity of heat Δq is added reversibly to a body at Kelvin temperture T then the increase of the entropy of the body is Δs defined by Δs=Δq/T.

2. A reversible process is one where the body is in equilibrium at every stage of the process. In this case a small change in conditions may reverse the process exactly.In a reversible process the gain of entropy by one body is exactly equal to the loss of entropy by the other and the entropy of the whole system is unchanged.

3. An irreversible process does not consist of equilibrium states at each stage of the process. An irreversible process creates entropy as the entropy gain by one body is greater than the entropy loss of the other.

4. Two objects each of heat capacity C are placed in thermal contact. The final equilibrium temperature of both objects is T. Initially the temperature of the "hot" object is T_h and the temperature of the "cold" object is T_c . Show that the change in entropy for the combined system is C ln(T²/(T_hT_c)). Assuming that all of the heat energy lost by the hot object is transferred to the cold object, show that T_h = T + ΔT and T_c = T - ΔT. Hence show that the change in entropy for the system is C ln(T²/(T²-ΔT²)) and show that this is greater than 1. Use Δq = C Δθ where Δθ is the infinitesimal change in temperature. This problem is taken from Problems for Physics Students by K F Riley. The solution and additional notes on Boltzmann's constant can be found in the article by Jeffrey J Prentis in The Physics Teacher, vol 34, Oct 1996, pp 392-397.

IB Physics The Centrifuge

A tutorial sheet on the pressure difference in a rotating fluid providing a centripetal force, an often misunderstood concept.

1. In a centrifuge a column of fluid of density ρ rotates in a horizontal plane at an angular speed ⍵. Show that the pressure in the fluid P at a distance r from the axis of rotation is given by P(r) = P₀ +1/2 ρ ⍵²(r² - r₀²), where P₀ is the pressure at r = r₀.
2. Explain why a marble goes to the outside of the centrifuge and a cork to the inside. See The Physics Teacher, vol 34, October 1996, p 422, The Marble, Cork and Centrifuge by Thomas J Pickett.
3. If the object is denser than the fluid the pressure in the fluid is not great enough to maintain the circular motion at that distance and so the object will drift outward through the fluid to a larger value of r.

IB Physics Gravitational Force Between Objects

This tutorial sheet is on the gravitational force between two objects. The first 3 questions are SL/HL and the others are HL.

1. Two uniform solid spheres of masses M₁ and M₂ and radii R₁ and R₂ have their centres a distance r apart (r > R₁ + R₂). State the magnitude of gravitational force acting on each mass.
2. A satellite of mass m orbits a planet of mass M in a circular path of radius r with orbital period T. Find an expression for the gravitational field strength at the position of the satellite.
3. Two point masses m and M are placed at rest at a distance d apart. After release the masses move towards each other. Find an expression for the speed of the smaller mass when the speed of the larger mass is V.
4. A uniform spherical shell has a mass M and a radius R. i. What is the gravitational field strength at the centre of the shell? ii. A particle is placed at the centre of the shell and given an initial speed u. Find the time taken to reach the shell.
5. A uniform rod of mass M and length L is in the same straight line as another uniform rod of mass M and length L. The distance between the centres of the rods is 2L. i. Can we find the attraction between the rods by considering each rod to be a point particle of mass M at its centre of mass? ii. Show that the attraction between the rods is GM²/L² ln(4/3)

IB Physics Magnetic flux through a loop

This tutorial sheet studies the magnetic flux through a circular loop. This topic was a HL Paper 2 question in Zone 3 in May 2025.

1. What is the meaning of the term magnetic flux?

2. A constant current I flows in a long straight wire. Write down an expression for the magnetic field at a perpendicular distance r from the wire.

3. A small circular loop of radius R is in the plane of the wire with its centre at a distance r from the wire. Write down an expression for the magnetic flux through the loop. What assumption are we making?

4. If I=2.0 A, r=10.0 cm and R = 2.0 cm show that the magnetic flux through the loop is approximately 2.5x 10^-8 Wb. This assumes that a constant magnetic field acts through the coil. In a show that question all substitution, rearranging and cancellations must be shown. Give the answer to one more decimal place to show evidence of calculator usage.

5. In question 4 determine the magnetic flux through the coil if the variation of field strength through the coil with distance from the straight wire is included. In this case there is slightly more flux through the coil since the magnetic field is stronger in the side of the loop closest to the straight wire. Flux = 2.54 x 10^-8 Wb.

IB Physics Where does the Sun's energy come from?

A tutorial work sheet on the topic of the source of the Sun’s energy. As this is a misunderstood concept, two key study points are taken from the article by E G Thomas, What’s so special about E = m c2, separating the myth from the truth, Eur. J. Phys., 26, (2005), S125. These are SL/HL concepts.

1. The Sun is getting its energy by rearranging protons into a more stable form of a helium nucleus. In this process the energy that is released comes from the binding energy of the helium nucleus.
2. The Sun does not turn matter into glowing energy.

IB Physics Mass and Energy

A tutorial sheet on mass and energy. Some study points are listed. These are taken from the article by Keith Atkin in Physics Education 35, 5, Sep 2000, page 319. These are SL/HL concepts.

1. Is matter the same as mass? No, matter is a qualitative term meaning substance, mass is a defined physical quantity which is a measure of inertia.
2. E = mc² states that mass (inertia) and energy are equivalent not interconvertible.
3. In the Cockroft-Walton reaction a proton interacts with a lithium nucleus forming two helium nuclei with a gain in kinetic energy. Is there a loss in mass? No, mass and energy are equivalent. Both energy and mass are conserved in the process, there is no destruction of mass or creation of energy. There is merely a redistribution of a fixed amount of energy (and its equivalent mass).

IB Physics Diffraction

This tutorial sheet is about diffraction, a concept that is sometimes misunderstood.

1. State the definition of diffraction.
2. Why does diffraction occur when light passes through a single slit?
3. Explain qualitatively the pattern that we see when light passes through a single slit if i. the slit width is much greater than the wavelength ii. the slit width is much less than the wavelength.
4. Divide a single slit of width b into two sections. By considering a ray from the top of each section, show that destructive interference occurs when b sin𝜃 = 𝜆. Do this with four sections in the slit and show that destructive interference occurs when b sin𝜃 = 2𝜆. This can be generalized to b sin𝜃 = n𝜆, n=1,2,..
5. When light is now incident normally on a diffraction grating what does the equation d sin𝜃 = n𝜆, n=0,1,2,3.. predict?
6. The incident monochromatic light is not normal to the grating. Can we use the equation d sin𝜃 = n𝜆 in this situation?
7. In question 6 the angle of incidence of the light on the grating is ϕ. Show that the angle 𝜃 to the normal to the grating where intensity maxima occur are given by d ( sin(𝜃 - ϕ) - sinϕ ) = n𝜆.
8. Describe the difference between primary maxima and secondary maxima.
9. Describe the interference pattern when white light is incident normally on a diffraction grating.