Booklet 1B - Cosmology (Worked Examples)

Question 1

Answer 1

Question 2

An arrow is fired at 400 ms^-1 at 30° to horizontal.

Calculate the horizontal and vertical components of the initial velocity.

Answer 2

The horizontal component of the initial velocity is given by:
u_h = 400 cos30
= 346 ms^-1

The vertical component is given by:
u_v = 400 sin30
= 200 ms^-1

Question 3

A projectile has the initial velocity components:
Horizontal: 346 ms^-1
Vertical: 200 ms^-1

Calculate

(a) the time of flight
(b) the greatest height reached
(c) the horizontal range

Answer 3

(a) for the whole flight,
u = 200 ms^-1
a = -9.8 ms^-2
s = 0 m
(landing at the same height it took off)
t = ?

s = ut + ½at²
0 = 200t + ½ x (-9.8) x t²
200t - 4.9t² = 0
4.9t = 200
t = 40.8 s

(b) at the greatest height, v = 0 ms^-1
v² = u² + 2as
0² = 200² + 2 x (-9.8) x s
s = 2040 m

(c) for horizontal motion,
t = 40.8 s
v = 346 ms-1

s = vt
s = 346 x 40.8
s = 14100 m

Question 4

A projectile has the initial velocity components:
Horizontal: 346 ms^-1
Vertical: 200 ms^-1

Calculate the height of the projectile after 5 s.

Answer 4

Question 5

A projectile has the initial velocity components:
Horizontal: 346 ms^-1
Vertical: 200 ms^-1

Calculate the (resultant) velocity of the projectile after 10 s of flight

Answer 5

Question 6

Consider a folder, of mass 0.3 kg and a pen, of mass 0.05 kg, sitting on a desk, 0.25
m apart. Calculate the magnitude of the gravitational force between the two masses.
Assume they can be approximated to spherical objects.

Answer 6

m<sub>1</sub> = 0.3 kg
m<sub>2</sub> = 0.05 kg
r = 0.25 m
G = 6.67 x 10<sup>-11</sup> m3 kg<sup>−1</sup> s<sup>−2</sup>
F = ?

F = Gm<sub>1</sub>m<sub>2</sub> / r<sup>2</sup>
F = 6.67 x 10<sup>-11</sup> x 0.3 x 0.05 / 0.25<sup>2</sup>
F = 1.6 x 10<sup>-11</sup> N

Question 7

A rocket is travelling past Earth at a constant speed of 2.7 × 10⁸ m s^–1.
The pilot measures the journey as taking 240 minutes.
How long did the journey take when measured by an observer on Earth?

Answer 7

Question 8

A spaceship is flying away from Earth towards Proxima Centauri at 0.8c. The distance to
Proxima Centauri is 4.2 ly. What does the pilot of the spaceship see this distance as?

Answer 8

v = 0.8c
l = 4.2 ly
l' = ?

Question 9

A girl tries out an experiment to illustrate the Doppler effect by spinning a battery operated
siren around her head at a constant speed of 5.0 ms^-1. The siren emits sound
waves with a frequency of 1200 Hz.
Describe what would be heard by a stationary observer standing a few metres away.

Answer 9

An observer a few metres away would hear the siren rise and fall in pitch with each rotation.

v_s = 5.0 ms^-1
f_s = 1200 Hz
v = 340 ms^-1 (speed of sound)
f_o = ?
As the siren is approaching the observer the frequency would be higher:
𝑓_o = 𝑓_𝑠 (𝑣/𝑣 − 𝑣𝑠)
𝑓_o = 1200 (340/340 − 5)
𝑓_o = 1218 𝐻𝑧
As the siren is moving away from the observer the frequency would be lower:
𝑓_o = 𝑓_𝑠 (𝑣/𝑣 + 𝑣𝑠)
𝑓_o = 1200 (340/340 + 5)
𝑓_o = 1183 𝐻𝑧

Question 10

The brightest spectral line for Sodium is measured at 589.0 nm on Earth. When the
spectrum from a distant galaxy is studied the equivalent line is found at 601.2 nm.
Calculate
a) The redshift
b) The speed at which the galaxy is moving away from us.

Answer 10

Question 11

A distant galaxy is found to have a speed of recession of 6.2 × 10⁶ ms^-1.

Calculate how far away it is.

Answer 11