# Projectile motion calculator

> Projectile motion calculator: range, maximum height, time of flight and impact speed for any launch angle and height, on Earth or other worlds, with drag.

انٹرایکٹو صورت: https://www.calcopenly.com/ur/science/projectile-motion-calculator
موضوع: سائنس کے کیلکولیٹر

The launch velocity splits into a horizontal part, v cos θ, which stays constant, and a vertical part, v sin θ, which gravity reduces by g every second. The time of flight is when the height returns to the ground, found by solving the vertical motion; the range is horizontal speed × flight time. On level ground this reduces to R = v² sin 2θ ÷ g.

It answers homework problems and sports or water-jet estimates. The default throw, 20 m/s at 45° on Earth, peaks at 10.20 m and lands 40.79 m away after 2.88 s. The same throw on the Moon, where g = 1.62 m/s², travels 246.9 m.

Without the drag option, air resistance, spin and the Earth's curvature are ignored, which overstates the range of fast or light objects. The linear-drag option shows how drag shortens and steepens the path; real balls at sports speeds meet drag closer to v², so read that result as a trend.

## اندراجات

- **Launch speed**
- **Launch angle above horizontal**
- **Launch height above ground**
- **Gravity** (اختیارات: Earth, Moon, Mars, Jupiter, Custom)
- **Gravitational acceleration**
- **نتائج کی اکائی** (اختیارات: Metres and m/s, Feet and mph)
- **Include air drag (linear model)**
- **Projectile mass**
- **Drag coefficient b (force = −b·v)**

## نتائج

- Horizontal range (m) — بنیادی نتیجہ
- Maximum height above ground (m)
- Time of flight (s)
- Time to maximum height (s)
- Impact speed (m/s)
- Impact angle below horizontal (°)
- Range without drag (m)

## فارمولا

$$
R = v_x T,\quad T = \frac{v_y + \sqrt{v_y^2 + 2gh_0}}{g},\quad H = h_0 + \frac{v_y^2}{2g};\qquad h_0 = 0:\ R = \frac{v^2\sin 2\theta}{g}
$$

## حل شدہ مثالیں

### 20 m/s at 45° on Earth

- Launch speed: 20 m/s
- Launch angle above horizontal: 45 °
- Launch height above ground: 0 m
- Gravity: Earth
- نتائج کی اکائی: Metres and m/s
- **Horizontal range: 40.7886 m**
- **Maximum height above ground: 10.1972 m**
- **Time of flight: 2.88419 s**
- **Impact speed: 20 m/s**
- جانچ کا ماخذ: ⁨Python 3.8 math: R = v²sin2θ/g, H = v²sin²θ/(2g), T = 2v·sinθ/g with g = 9.80665⁩

### 15 m/s at 30° from a 10 m cliff

- Launch speed: 15 m/s
- Launch angle above horizontal: 30 °
- Launch height above ground: 10 m
- Gravity: Earth
- نتائج کی اکائی: Metres and m/s
- **Horizontal range: 30.979 m**
- **Maximum height above ground: 12.868 m**
- **Time of flight: 2.38477 s**
- **Impact speed: 20.5215 m/s**
- **Impact angle below horizontal: 50.73 °**
- جانچ کا ماخذ: ⁨Python 3.8 math: T = (v_y + √(v_y² + 2gh))/g = 2.3847661…, R = v_x·T, v_impact = √(v² + 2gh)⁩

### Same throw on the Moon

- Launch speed: 20 m/s
- Launch angle above horizontal: 45 °
- Launch height above ground: 0 m
- Gravity: Moon
- نتائج کی اکائی: Metres and m/s
- **Horizontal range: 246.914 m**
- **Time of flight: 17.4594 s**
- جانچ کا ماخذ: ⁨Python 3.8 math: 400/1.62 and 2·20·sin45°/1.62 (NASA g_Moon = 1.62 m/s²)⁩

### Horizontal launch from 20 m (θ = 0)

- Launch speed: 10 m/s
- Launch angle above horizontal: 0 °
- Launch height above ground: 20 m
- Gravity: Earth
- نتائج کی اکائی: Metres and m/s
- **Horizontal range: 20.1962 m**
- **Maximum height above ground: 20 m**
- **Time of flight: 2.01962 s**
- **Impact angle below horizontal: 63.21 °**
- جانچ کا ماخذ: ⁨Python 3.8 math: T = √(2h/g) = 2.0196200…, R = vT, tan φ = gT/v⁩

### Straight up (θ = 90°): zero range

- Launch speed: 10 m/s
- Launch angle above horizontal: 90 °
- Launch height above ground: 0 m
- Gravity: Earth
- نتائج کی اکائی: Metres and m/s
- **Horizontal range: 0 m**
- **Maximum height above ground: 5.09858 m**
- **Time of flight: 2.03943 s**
- جانچ کا ماخذ: ⁨Python 3.8 math: H = v²/(2g) = 5.0985811, T = 2v/g = 2.0394324⁩

### Linear drag, 0.145 kg ball, b = 0.01 kg/s

- Launch speed: 20 m/s
- Launch angle above horizontal: 45 °
- Launch height above ground: 0 m
- Gravity: Earth
- نتائج کی اکائی: Metres and m/s
- Include air drag (linear model): ہاں
- Projectile mass: 0.145 kg
- Drag coefficient b (force = −b·v): 0.01 kg/s
- **Horizontal range: 35.9451 m**
- **Maximum height above ground: 9.56778 m**
- **Time of flight: 2.79449 s**
- جانچ کا ماخذ: ⁨Python 3.8 floats: closed-form x(t), y(t) for F = −bv (Taylor §2.4), landing time by Newton iteration⁩

## سوالات

### What launch angle gives the maximum range?

On level ground with no air resistance, 45° gives the longest range, because R = v² sin 2θ ÷ g and sin 2θ peaks when 2θ = 90°. Launching from a height moves the best angle lower: from a 10 m cliff at 15 m/s it is about 36.2°, which reaches 31.4 m against 30.5 m at 45°. Air drag also lowers the best angle.

### How do you calculate the time of flight?

On level ground, T = 2v sin θ ÷ g, so a 20 m/s launch at 45° on Earth stays up 2.88 s. From a launch height h, take the positive root of h + v sin θ·t − ½gt² = 0, which is T = (v sin θ + √(v² sin² θ + 2gh)) ÷ g. The time to the highest point is v sin θ ÷ g, half the level-ground flight time.

### Why do 30° and 60° give the same range?

Complementary angles give the same range on level ground because sin 2θ = sin(180° − 2θ). At 20 m/s, launches at 30° and 60° both land 35.32 m away. The 60° shot climbs three times as high, 15.30 m against 5.10 m, and stays up 3.53 s instead of 2.04 s, which matters when the path has to clear an obstacle.

### How much does air resistance shorten the range?

It depends on the object's mass, size and speed. With the linear model here, a 0.145 kg ball thrown at 20 m/s and 45° with b = 0.01 kg/s lands 35.95 m away instead of 40.79 m, about 12% shorter, and peaks at 9.57 m instead of 10.20 m. Drag also makes the descent steeper than the climb, so the path is no longer a symmetric parabola.

### “⁨Projectile motion calculator⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 6۔ مثلاً، “⁨20 m/s at 45° on Earth⁩” کو ⁨Python 3.8 math: R = v²sin2θ/g, H = v²sin²θ/(2g), T = 2v·sinθ/g with g = 9.80665⁩ سے جانچا جاتا ہے۔

### اس طریقے کا ماخذ کیا ہے؟

OpenStax University Physics Volume 1, §4.3 Projectile motion; Taylor, Classical Mechanics (2005), §2.2–2.3 Linear air resistance; NASA Planetary Fact Sheet — surface gravity.

## ماخذ

- [OpenStax University Physics Volume 1, §4.3 Projectile motion](https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion)
- Taylor, Classical Mechanics (2005), §2.2–2.3 Linear air resistance
- [NASA Planetary Fact Sheet — surface gravity](https://nssdc.gsfc.nasa.gov/planetary/factsheet/)