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Practice sheet · Space Path · Lens 1: Math

Big numbers & scale

Practice for The Math of Space: notation, scale, light-time, orbits, gravity, Kepler’s law and meteor counts (pp.8–16), plus the Algebra Level Up (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Scientific notation: one digit before the decimal point, times a power of ten. 384,400 = 3.844 × 10⁵. To multiply, multiply the front numbers and add the powers; to divide, divide and subtract.

Light: about 299,792 km/s (300,000 for quick sums). time = distance ÷ speed. 1 AU ≈ 1.496 × 10⁸ km; 1 light-year ≈ 9.461 × 10¹² km (p.8).

Warm-up

One step

Writing and combining powers of ten.
1

Write it short

Write in scientific notation: Mars’s diameter, 6,779 km, and its average distance from the Sun, 227,900,000 km.

2

Write it long

Write as ordinary numbers: 5.2 × 10⁴ and 3 × 10⁻³.

3

Multiply

Work out (4 × 10⁵) × (2 × 10³) and (5 × 10⁴) × (3 × 10²). Give both in scientific notation.

4

Divide

Work out (9 × 10⁸) ÷ (3 × 10²).

Core

Two or three steps

Keep the units with every number.
5

Light delay

How long does light take to reach the Moon (384,400 km)? To reach Mars at its closest, about 55 million km? Give seconds, then minutes for Mars.

6

Model Jupiter

In the 30 cm-Sun model (p.9), the scale is 0.30 m ÷ 1,391,400,000 m. Jupiter is about 139,820 km across and 5.2 AU from the Sun. How big is model Jupiter, how far is 1 AU, and where does Jupiter go?

Notation Practice · Space Path · Unizon1/2
Space Path · Unizon · PracticeScientific notation & scale
7

A higher orbit

A satellite orbits 600 km above Earth (radius 6,371 km). How long is one lap? One lap takes about 96.5 minutes: how fast is it going in km/h?

8

Space scale

A backpack reads 40 on an Earth bathroom scale. What would it read on the Moon (0.165) and Mars (0.38)? A reading of 101.2 on Jupiter (2.53) is what on Earth?

9

To Sirius

Sirius, the brightest star in the night sky, is about 8.6 light-years away. How far is that in km, in scientific notation?

10

Meteor count

Five 15-minute counts during a shower: 14, 10, 17, 11, 13. Find the mean, the median and the range. What rate per hour does the mean give?

Level Up

Laws and equations

Kepler (p.15) and A1–A2 (pp.17–18).
11

Venus’s year

Use Kepler’s law T = a1.5 (p.15). Venus is 0.723 AU from the Sun. How long is its year, in Earth years and in days?

12

Run Kepler backwards

Uranus takes about 84 Earth years to orbit. Solve T² = a³ for a: a = T2/3. How far out is it, in AU?

13

Ping Jupiter

Use d = 300,000t (p.17). A signal from a spacecraft near Jupiter takes 43 minutes to arrive. How far away is it? Write it in scientific notation.

14

Moon toss

On the Moon, a ball thrown up at 6 m/s has height h = 6t − 0.8t² (p.18 model). When does it land? When is it 10 m up?

Make your own problem

Your turn to be the mission planner

  1. Pick two objects from the book: a planet, a moon, a star, a spacecraft.
  2. Write a distance, light-time or scale-model question using scientific notation.
  3. Solve it, then check the size of the answer: is it sensible?

My problem, and my worked answer:

Notation Practice · Space Path · Unizon2/2
Space Path · Unizon · PracticeScientific notation & scale

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

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Answer key

Worked solutions. Distances are averages; small rounding differences are fine.

Warm-up
1
6.779 × 10³ km · 2.279 × 10⁸ km Move the decimal point 3 places, then 8 places.
2
52,000 · 0.003 × 10⁴ moves the point 4 places right; × 10⁻³ moves it 3 places left.
3
8 × 10⁸ · 1.5 × 10⁷ 4 × 2 = 8 and 5 + 3 = 8. 5 × 3 = 15 and 4 + 2 = 6: 15 × 10⁶ = 1.5 × 10⁷.
4
3 × 10⁶ 9 ÷ 3 = 3 and 8 − 2 = 6.
Core
5
≈ 1.28 s · ≈ 183 s ≈ 3.1 min 384,400 ÷ 299,792 ≈ 1.28 s. 55,000,000 ÷ 299,792 ≈ 183 s ≈ 3.1 min. At its farthest Mars is several times this.
6
≈ 3.0 cm · ≈ 32.3 m · ≈ 168 m 139,820,000 m × scale ≈ 0.030 m. 1 AU: 149,600,000,000 m × scale ≈ 32.3 m (p.9 rounds to 32). 5.2 × 32.3 ≈ 168 m away: walnut-sized, a long field away.
7
≈ 43,800 km · ≈ 27,200 km/h r = 6,371 + 600 = 6,971 km. 2π × 6,971 ≈ 43,800 km. 96.5 min ≈ 1.61 h; 43,800 ÷ 1.61 ≈ 27,200 km/h, a bit slower than the ISS (higher is slower, p.11).
8
6.6 · 15.2 · 40 40 × 0.165 = 6.6; 40 × 0.38 = 15.2; 101.2 ÷ 2.53 = 40. The mass is the same everywhere.
9
≈ 8.1 × 10¹³ km 8.6 × 9.461 × 10¹² ≈ 81.4 × 10¹² = 8.14 × 10¹³ km.
10
mean 13 · median 13 · range 7 · 52/h 65 ÷ 5 = 13. In order 10, 11, 13, 14, 17: the middle is 13. 17 − 10 = 7. 13 × 4 = 52 an hour.
Level Up
11
≈ 0.615 yr ≈ 225 days 0.7231.5 ≈ 0.615 years × 365.25 ≈ 225 days (the real year is about 224.7 days).
12
≈ 19.2 AU 842/3 ≈ 19.2. Check: 19.2³ ≈ 7,078 and 84² = 7,056.
13
7.74 × 10⁸ km 43 min = 2,580 s. 300,000 × 2,580 = 774,000,000 km.
14
lands at 7.5 s · 10 m at 2.5 s and 5 s 6t − 0.8t² = 0 → t(6 − 0.8t) = 0 → t = 7.5. 0.8t² − 6t + 10 = 0 → t² − 7.5t + 12.5 = 0 → t = (7.5 ± 2.5) ÷ 2.
Make your own: write every big number in scientific notation, keep units on every line, and compare your answer with something you know (the Moon is about 1.3 light-seconds away).
Notation Practice · Space Path · UnizonAnswer key