Tool 3: Mathematics (DP IB Physics: HL): Exam Questions

A stone falls from rest to the bottom of a water well of depth d. The time t taken to fall is 3.0 ± 0.3 s. The depth of the well is calculated to be 30 m using the equation:

The uncertainty in a is negligible.

What is the absolute uncertainty in d?

Three forces act on a body as shown.

Which fourth force is required so that the resultant force is zero?

Velocity is a vector quantity, so it can be represented by a vector arrow.

Which physical quantity is represented by the length of its vector arrow?

A student collects values of the time period, T, of a pendulum at different lengths of string, L. They want to investigate the relationship:

They plot the values on a graph along with the error bars associated with each point, as shown in the diagram.

Which of the following is the closest approximation of the percentage uncertainty in the experimental value of g?

The diagram shows an experiment to measure the speed of a small ball falling at constant speed through a clear liquid in a glass tube.

There are two marks on the tube. The top mark is positioned at 15 ± 1 mm on the adjacent rule and the lower mark at 285 ± 1 mm. The ball passes the top mark at 2.50 ± 0.02 s and passes the lower mark at 4.50 ± 0.02 s.

The constant speed of the ball is calculated to be 135 mm s^–1.

Which expression calculates the fractional uncertainty in the value of this speed?

A rectangular object sits at rest on a plane inclined at angle  to the horizontal.

R is the normal force, W is the weight and F is friction. 

Which row correctly labels R and F in terms of mass m and acceleration due to gravity g.

Which of the following represents correct vector and scalar quantities?

The strain energy W of a spring is determined from its spring constant k and extension x. The spring obeys Hooke’s law, and the value of W is calculated using the equation shown.

The spring constant k is 200 ± 1 N m^–1 and the extension x is 0.040 ± 0.004 m.

What is the percentage uncertainty in the calculated value of W?

In an experiment investigating the electrolysis of copper, a student sets out to find the electrochemical equivalent, Z.

The electrochemical equivalent of a substance is the amount of substance deposited on a cathode per Coulomb of charge.

This can be determined using the equation:

Where:

• Mass of cathode before passing current, = (54.39 ± 0.01) × 10^–3 kg

• Mass of cathode after passing current, = (52.06 ± 0.01) × 10^–3 kg

• Current, = 3.00 ± 1 A

• Time, t = 4800 ± 100 s

What is the largest possible value of Z from these readings?

The sides of a square are measured to be 8.0 ± 0.2 cm.

Which of the following gives the area of the square and its uncertainty?

Blue light has a wavelength of 450 nm.

Which of the following also represents this wavelength?

In an experiment, a radio-controlled car takes 1.50 ± 0.05 s to travel 30.0 ± 0.1 m.

What is the car’s average speed and the uncertainty in this value?

The equation of hydrostatic pressure relates the pressure of a fluid P to the density of the fluid ρ, and is given by:

where g is the gravitational field strength, and Δh is the depth from the surface of the fluid.

How can both sides of this equation be written in terms of SI base units?

The diagram shows the initial and final readings on a thermometer before and after a liquid is heated.

What is the best estimate for the temperature increase of the liquid?

A glider is momentarily stationary in the air before losing 1.00 m of vertical height for every 6.00 m that it goes forward horizontally. The horizontal speed of the glider is 12 m s^–1.

What is the vertical speed of the glider, to an appropriate number of significant figures, after it falls a total of 31.25 m?

Which of the following represents the correct values of the x-component and y-component of the vector -F ?

An object falls for a time of 0.081 s. The acceleration of free fall is 9.81 m s^–2. The displacement is calculated using these values.

Which of the following gives the correct number of significant digits for the calculated value of the displacement of the object?

The magnitude of a is 15 N, and the magnitude of b is 30 N.

Which of the following represents the correct resultant horizontal and vertical components of the vectors in the diagram?

The diagram shows vector p.

In which of the following diagrams is vector p multiplied by a scalar represented?

In which of the following diagrams is the addition of vectors x and y represented?

In which of the following diagrams is q – p represented correctly?

The arrow represents the vector R.

Which diagram does not represent R as two perpendicular components?

The table contains some quantities, together with their symbols and units.

Which expression has the units of energy?

For which quantity is the magnitude a reasonable estimate?

0.9 kilowatt-hours (kW h) is equivalent to

The units of all physical quantities can be expressed in terms of SI base units.

 Which pair contains quantities with different base units?

The best estimate for the time it takes light to cross the nucleus of a hydrogen atom is:

The graph shows the data for the variation of the power output of a photovoltaic cell with load resistance. The data were obtained by placing the cell in sunlight. The intensity of the energy from the Sun incident on the surface of the cell was constant.

The intensity of the Sun’s radiation incident on the cell is 0.6 W m^–2. The active area of the cell has dimensions of 60 cm × 60 cm.

At the peak power, what is the ratio ?

Which of the following estimates is realistic?

A boy jumps from a wall 1.25 metres high.

What is the best estimate for the boy's change in momentum when he lands on the ground without rebounding?

When a constant braking force is applied to a vehicle moving at speed v, the distance d moved by the vehicle as it comes to rest is given by the expression:

In this equation, k is a constant.

When d is measured in metres and v is measured in metres per second, the constant has a value of k₁.

What is the value of the constant when the distance is measured in metres, and the speed is measured in kilometres per hour?

What is the unit for the gas constant in fundamental SI units?

Which of the following gives the correct unit for ?

The frictional force F on a sphere falling through a fluid is given by the formula:

In this equation, a is the radius of the sphere, η is a constant relating to the fluid and v is the velocity of the sphere.

What are the units of η?

The speed v of a liquid leaving a tube depends on the change in pressure ΔP and the density ρ of the liquid. The speed is given by the equation:

In this equation, k is a constant that has no units.

What is the value of n?

The theory of gas flow through small diameter tubes at low pressures is an important consideration of high vacuum techniques.

One equation used in this theory is:

Where k is a unitless constant, r is the radius of the tube, p₁ and p₂ are the pressures at each end of the tube, L is the length of the tube, M is the molar mass of the gas, R is the molar gas constant and T is the thermodynamic temperature of the gas.

What are the base units of Q?

What is the correct value for the gravitational constant ?

A small conker is attached to a string and whirled overhead in a horizontal circle. The tension in the string T is measured for different values of linear speed v, and the following graph is obtained.

These experimental results suggest that there is...

The drag coefficient C_d is a number with no units. It is used to compare the drag on different cars at different speeds. It is given by the equation:

F is the drag force on the car, ρ is the density of the air, A is the cross-sectional area of the car and v is the speed of the car.

 What is the value of n?

The following processes are given: 

I. The friction required to bring a car to a complete halt. 

II. The effort required to nail a hammer into wood. 

III. The heat required to cook. 

Which of the processes involves a scalar quantity? 

A drifting spider of mass 50 g hangs on its web from point P on a ceiling. It is being displaced at an angle θ to the vertical by a gentle breeze equivalent to 0.2 N. 

What is the magnitude of the angle θ?

A researcher plots a graph to investigate the variation of the current and potential difference of a resistor.

After realising they have read the manufacturer's label of their electrical equipment incorrectly, they determine that the size of the error bars shown should decrease.

Which row correctly describes how this will affect the value of the maximum gradient G_max, the minimum gradient G_min, and the resistance of the resistor R?

A horizontal force accelerates a body of mass up a rough plane inclined at an angle to the horizontal. The coefficient of dynamic friction between the surface and the body is .

Which expression describes the magnitude of the resultant force on the body?  

A box rests on an inclined plane. 

Which of the following statements is correct if the incline becomes less steep? 

Two lengths, n and n – 2 have an uncertainty of . The uncertainty  is much less than n. 

In which of the following quantities is the percentage uncertainty the largest? 

In an experiment to determine the acceleration of free fall g, a student presents their calculated value as 9.7835 m s^–2 and states their value is only accurate to ± 5%.

Which of the following gives their calculated value to an appropriate number of significant figures? 

The length and breadth of a rectangular card were found to be (5.0 ± 0.2) cm and (3.0 ± 0.2) cm, respectively.

The area of the card cannot be more than

An object dropped from the edge of a cliff is timed to take (10.0 ± 0.1) s to fall to the surface of the sea.

If the acceleration due to free fall is (10.0 ± 0.2) m s^–2, the calculated height of the cliff from which the object was released should be quoted as: 

A pilot wishes to set her plane on a course such that it flies directly west of its starting point. She must take into account the strong northerly wind, meaning that she is flying in a wind which originates in the north and blows in a southward direction. 

On which course should the plane be directed? 

A positron is in the effect of two electric fields, shown measured in arbitrary units. 

What is the magnitude of the resultant field strength on the positron?

Alice sits on the platform of a train station, observing the trains as they move past, from left to right, at constant velocity. 

Bob is sat in one particular train carriage, and throws a tennis ball vertically upward, catching it as it comes back down.

Which diagram best represents the trajectory of the tennis ball relative to Alice? (Note: this is the path the ball will follow relative to Alice's position, not what she will see.)

A car drives from its base camp, at position X, to lodge P, 40 km away, at a bearing of 030° from X. After dropping off supplies, it continues to lodge Q, which is 50 km away and at a bearing of 330° from P. 

What is the distance between the base camp and lodge Q?