Section 1: Units, measurement and kinematics

Write the dimensions of every term before you trust a formula, and take a sign convention for direction before you write a single equation of motion.

A lesson for this section

Section A: multiple choice

Question 1Section A: multiple choice

Light of intensity

I

(power per unit area) falls normally on a square mirror of side

s

. The force on the mirror depends only on

I

, the mirror's area

A

and the speed of light

c

, as

F

= kI^{a}A^{b}c^{d}

with

k

a dimensionless constant. In an experiment

I

is found from this relation, with

F

measured to within

2\%

s

to within

0.25\%

, and

c

and

k

known exactly. Using dimensional analysis to find

a

b

and

d

, what is the maximum percentage error in

I

Question 2Section A: multiple choice

At high speed the drag force

F

on a sphere depends on the density

\rho

of the air, the sphere's speed

v

and its radius

r

F

= k\rho^{a} v^{b} r^{c}

, where

k

is a dimensionless constant. Which relation does dimensional analysis give?

Question 3Section A: multiple choice

Suppose force

F

, acceleration

A

and time

T

are chosen as the base quantities instead of mass, length and time. What are the dimensions of surface tension in this system?

Question 4Section A: multiple choice

In the relation

E

= \frac{\alpha - t^2}{\beta x}

E

is an energy,

t

is a time and

x

is a distance. What are the dimensions of

\frac{\alpha}{\beta}

Question 5Section A: multiple choice

A new system of units takes force, speed and time as base quantities, with units

10\ \mathrm{N}

4\ \mathrm{m\,s^{-1}}

and

2\ \mathrm{s}

. The Young's modulus of steel is

2 \times 10^{11}\ \mathrm{Pa}

. What is its numerical value in the new system?

Question 6Section A: multiple choice

In a simple model of an atom, an electron of mass

m

is attracted to the nucleus by a force

\frac{k}{r^2}

, where

r

is their separation and

k

= \frac{e^2}{4\pi\varepsilon_0}

; quantum effects bring in Planck's constant

h

. Which combination of

m

k

and

h

has the dimensions of length?

Question 7Section A: multiple choice

To find a liquid's density, a student weighs an empty beaker,

0.0503\ \mathrm{kg}

, and then the beaker holding

2.15 \times 10^{-6}\ \mathrm{m^3}

of the liquid,

0.05248\ \mathrm{kg}

. What density should be reported, to the correct number of significant figures?

Question 8Section A: multiple choice

A pendulum of length

(1.000 \pm 0.004)\ \mathrm{m}

is timed for

20

swings five times:

40.2\ \mathrm{s}

39.8\ \mathrm{s}

40.4\ \mathrm{s}

39.6\ \mathrm{s}

and

40.0\ \mathrm{s}

. The value of

g

is found from

g

= \frac{4\pi^2 L}{T^2}

, taking the error in the time as the mean absolute error. What is the percentage error in

g

Question 9Section A: multiple choice

A solid cylinder has mass

(0.400 \pm 0.002)\ \mathrm{kg}

, diameter

(20.0 \pm 0.1)\ \mathrm{mm}

and length

(40.0 \pm 0.2)\ \mathrm{mm}

. What is the maximum percentage error in its density?

Question 10Section A: multiple choice

A trolley of mass

(0.250 \pm 0.005)\ \mathrm{kg}

moves at a steady speed. It covers

(2.00 \pm 0.01)\ \mathrm{m}

(0.80 \pm 0.01)\ \mathrm{s}

. What is the maximum percentage error in the kinetic energy found from these readings?

Question 11Section A: multiple choice

On a vernier calliper the main scale is marked in

0.5\ \mathrm{mm}

divisions and

20

vernier divisions span

19

main-scale divisions. With the jaws closed, the vernier zero lies to the right of the main-scale zero and the 4th vernier division lines up with a main-scale mark. Gripping a rod, the vernier zero lies just past the

23.5\ \mathrm{mm}

mark and the 13th vernier division lines up. What is the rod's diameter?

Question 12Section A: multiple choice

A screw gauge has a pitch of

0.5\ \mathrm{mm}

and

50

divisions on its circular scale. With the faces closed it reads

-0.03\ \mathrm{mm}

. On a wire, the main scale shows

3

divisions of

0.5\ \mathrm{mm}

and the

37

th circular division is on the reference line. What is the wire's diameter?

Question 13Section A: multiple choice

A heater warms a liquid of mass

m

; the specific heat capacity is found from

c

= \frac{VIt}{m\,\Delta T}

. The voltage is known to

2\%

, the current to

1\%

, the mass to

0.5\%

and the time exactly. The thermometer reads

(298.0 \pm 0.2)\ \mathrm{K}

at the start and

(313.0 \pm 0.3)\ \mathrm{K}

at the end. What is the maximum percentage error in

c

Question 14Section A: multiple choice

A particle moves along a straight line with constant acceleration. It covers

12\ \mathrm{m}

during the third second of its motion and

4\ \mathrm{m}

during the fifth second, without turning round in either second. What distance does it cover during the sixth second?

Question 15Section A: multiple choice

The graph shows the velocity of a cyclist moving along a straight track from

t

= 0

t

= 9\ \mathrm{s}

. What is the cyclist's average speed over these

9\ \mathrm{s}

Question 16Section A: multiple choice

Car A moves at

20\ \mathrm{m\,s^{-1}}

along a straight road,

75\ \mathrm{m}

behind car B, which moves the same way at a steady

10\ \mathrm{m\,s^{-1}}

. At this instant A's driver brakes, giving A a constant deceleration of

1\ \mathrm{m\,s^{-2}}

. What is the smallest distance between the cars?

Question 17Section A: multiple choice

A particle moves along the

x

axis with acceleration

a

= (6t - 8)\ \mathrm{m\,s^{-2}}

, where

t

is in seconds. At

t

= 0

it is at

x

= 0

, moving at

4\ \mathrm{m\,s^{-1}}

in the

+x

direction. What distance does it travel between

t

= 0

and

t

= 3\ \mathrm{s}

Question 18Section A: multiple choice

A particle moves along the

+x

axis so that its velocity at position

x

v

= \frac{2}{x}

, with

v

\mathrm{m\,s^{-1}}

and

x

in metres. How long does it take to go from

x

= 1\ \mathrm{m}

x

= 4\ \mathrm{m}

Question 19Section A: multiple choice

Ball A is released from rest

60\ \mathrm{m}

above the ground. At the same instant ball B is thrown vertically upwards at

20\ \mathrm{m\,s^{-1}}

from the ground directly below. Take

g

= 10\ \mathrm{m\,s^{-2}}

. Where do they meet, and which way is B moving then?

Question 20Section A: multiple choice

A stone is dropped from rest into a well. The splash is heard

4.25\ \mathrm{s}

after release. Take

g

= 10\ \mathrm{m\,s^{-2}}

and the speed of sound as

320\ \mathrm{m\,s^{-1}}

. How deep is the water surface below the point of release?

Question 21Section A: multiple choice

A lift starts from rest and moves up with a constant acceleration of

5\ \mathrm{m\,s^{-2}}

. One second later a bolt comes loose from its ceiling,

2.7\ \mathrm{m}

above its floor. Take

g

= 10\ \mathrm{m\,s^{-2}}

. What is the bolt's displacement, measured from the ground, between coming loose and hitting the floor?

Question 22Section A: multiple choice

A stone is thrown from the edge of a cliff at

20\ \mathrm{m\,s^{-1}}

, at

30^\circ

above the horizontal. It hits the level ground below moving at

60^\circ

below the horizontal. Take

g

= 10\ \mathrm{m\,s^{-2}}

. How high is the cliff?

Question 23Section A: multiple choice

Two balls are projected from the same point on level ground with the same speed but at different angles, and both land at the same point

40\ \mathrm{m}

away. The greatest height of one flight is

15\ \mathrm{m}

more than that of the other. Take

g

= 10\ \mathrm{m\,s^{-2}}

. What is the speed of projection?

Question 24Section A: multiple choice

A train moves along a straight level track with a constant acceleration of

2\ \mathrm{m\,s^{-2}}

. A passenger standing still in the train throws a ball vertically upwards, relative to herself, at

10\ \mathrm{m\,s^{-1}}

. Take

g

= 10\ \mathrm{m\,s^{-2}}

and ignore air resistance. Where is the ball, compared with her hand, when it falls back to the height from which it was thrown?

Question 25Section A: multiple choice

A river

120\ \mathrm{m}

wide flows at

3\ \mathrm{m\,s^{-1}}

. A swimmer can swim at

5\ \mathrm{m\,s^{-1}}

relative to the water. How much longer does she take to cross if she lands directly opposite her starting point than if she crosses in the least possible time?

Question 26Section A: multiple choice

A man walking east at

2\ \mathrm{m\,s^{-1}}

sees the rain falling vertically. When he walks west at

2\ \mathrm{m\,s^{-1}}

, the rain appears to fall at

30^\circ

to the vertical. What is the actual speed of the rain?

Question 27Section A: multiple choice

A bead slides round a circular wire of radius

4\ \mathrm{m}

. Its speed falls with the distance

s

it has travelled along the wire as

v

= (10 - 2s)\ \mathrm{m\,s^{-1}}

, with

s

in metres. What is the magnitude of its acceleration when

s

= 1\ \mathrm{m}

Question 28Section A: multiple choice

A particle moves round a circle of radius

2\ \mathrm{m}

at a constant speed of

4\ \mathrm{m\,s^{-1}}

. What is the magnitude of its average acceleration over a quarter of a revolution?

Question 29Section A: multiple choice

A ball is projected from level ground with velocity

(6\hat{i} + 8\hat{j})\ \mathrm{m\,s^{-1}}

, where

\hat{j}

is vertically upwards. Take

g

= 10\ \mathrm{m\,s^{-2}}

. At what time after launch is its velocity perpendicular to its initial velocity?

Question 30Section A: multiple choice

A plane slope rises at

30^\circ

to the horizontal. A ball is projected from a point on the slope at

20\ \mathrm{m\,s^{-1}}

, at

60^\circ

above the horizontal, in the vertical plane up the line of greatest slope. Take

g

= 10\ \mathrm{m\,s^{-2}}

. How far up the slope does it land?

Section B: numerical value

Units, measurement and kinematicsSection B: numerical value · 4 marks

A new system of units takes length, speed and density as base quantities, with units

2\ \mathrm{m}

4\ \mathrm{m\,s^{-1}}

and

1000\ \mathrm{kg\,m^{-3}}

. What is the numerical value in this system of a pressure of

1.6 \times 10^{6}\ \mathrm{Pa}

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

The frequency

f

of small oscillations of a liquid drop depends only on its surface tension

\sigma

, its density

\rho

and its radius

r

. In an experiment

\sigma

is known to

2\%

\rho

1\%

and

r

1\%

. Using dimensional analysis to find how

f

depends on them, find the maximum percentage error in

f

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

A metal cube has side

(25.0 \pm 0.1)\ \mathrm{mm}

and mass

(0.0500 \pm 0.0001)\ \mathrm{kg}

. The maximum percentage error in its density is

\frac{x}{10}\%

. Find

x

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

A driver needs

0.5\ \mathrm{s}

to react before braking, and the brakes then give a constant deceleration of

5\ \mathrm{m\,s^{-2}}

. By how many metres is the total stopping distance from

90\ \mathrm{km\,h^{-1}}

longer than from

72\ \mathrm{km\,h^{-1}}

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

A balloon rises vertically at a steady

5\ \mathrm{m\,s^{-1}}

. When it is

30\ \mathrm{m}

above the ground, a packet is let go from it. Take

g

= 10\ \mathrm{m\,s^{-2}}

and ignore air resistance on the packet. How many metres above the ground is the balloon when the packet lands?

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

Ball A is thrown vertically upwards at

20\ \mathrm{m\,s^{-1}}

from a point O on level ground. At the same instant ball B is thrown horizontally at

15\ \mathrm{m\,s^{-1}}

, directly towards the vertical line through O, from a point

40\ \mathrm{m}

above the ground and

30\ \mathrm{m}

from that line. Take

g

= 10\ \mathrm{m\,s^{-2}}

. The balls collide. At what height in metres above the ground?

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

A ball can be projected from level ground at

20\ \mathrm{m\,s^{-1}}

at either of two launch angles so that it lands

20\sqrt{3}\ \mathrm{m}

away on the same level. Take

g

= 10\ \mathrm{m\,s^{-2}}

. Find the sum, in metres, of the greatest heights reached in the two flights.

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Units, measurement and kinematicsSection B: numerical value · 4 marks

A particle starts from rest on a circle of radius

8\ \mathrm{m}

and moves round it with speed

v

= 2t

, where

v

is in

\mathrm{m\,s^{-1}}

and

t

in seconds. How many metres has it travelled when its acceleration makes an angle of

45^\circ

with its velocity?

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.