Quick Answer – What is Projectile Motion?
Projectile motion is the motion of an object thrown into the air and moving under the influence of gravity alone (neglecting air resistance). It follows a parabolic path, and its key parameters — maximum height, time of flight, and range — depend only on the initial velocity and the angle of projection.
Theory & Formulas
For a projectile launched from the ground with initial speed u at an angle θ to the horizontal, assuming acceleration due to gravity g = 9.8 m/s² and no air resistance, the following equations hold:
Maximum Height H = (u² sin²θ) / (2g)
Time of Flight T = (2u sinθ) / g
Range R = (u² sin 2θ) / g
Here θ is measured from the horizontal. The maximum range occurs at θ = 45° because sin(90°) = 1. For complementary angles (e.g., 30° and 60°), the range is the same, but the maximum height is greater for the larger angle.
Solved Examples
Example 1: Typical JEE Problem
Given: u = 20 m/s, θ = 30°. Find H, T, R.
sin(30°) = 0.5, sin²(30°) = 0.25
H = (20² × 0.25) / (2 × 9.8) = (400 × 0.25) / 19.6 = 100 / 19.6 ≈ 5.10 m
T = (2 × 20 × 0.5) / 9.8 = 20 / 9.8 ≈ 2.04 s
sin(2 × 30°) = sin(60°) ≈ 0.8660
R = (20² × 0.8660) / 9.8 = (400 × 0.8660) / 9.8 ≈ 35.35 m
Example 2: Angle of Maximum Range
Given: u = 15 m/s, θ = 45°. Find R.
sin(2 × 45°) = sin(90°) = 1
R = (15² × 1) / 9.8 = 225 / 9.8 ≈ 22.96 m (maximum for this speed)
Example 3: Vertical Throw
Given: u = 10 m/s, θ = 90°. Find H, T.
sin(90°) = 1, so H = (10² × 1) / (2 × 9.8) = 100 / 19.6 ≈ 5.10 m
T = (2 × 10 × 1) / 9.8 = 20 / 9.8 ≈ 2.04 s
Range R = 0 because sin(180°) = 0
Frequently Asked Questions
Q1. What units should I use for the projectile motion calculator?
Enter velocity in metres per second (m/s) and the angle in degrees. The tool automatically converts the angle to radians for trigonometric calculations. All results are in SI units: height and range in metres, time in seconds. Gravity is fixed at 9.8 m/s².
Q2. Can I enter decimal values for velocity and angle?
Yes. You can use decimals like 15.5 m/s or 37.5°. The calculator accepts both dot and comma as decimal separators (e.g., 15,5 is automatically treated as 15.5). Scientific notation (e.g., 1e2 for 100) is also supported.
Q3. Why is the acceleration due to gravity taken as 9.8 m/s²?
This is the standard value used in NCERT textbooks and most JEE problems, representing the approximate gravitational acceleration near Earth's surface. If your problem uses g = 10 m/s², you can mentally adjust the results by multiplying by 9.8/10 or re‑input a scaled velocity, but for precise work, the calculator always uses 9.8.
Q4. What happens if I enter a negative velocity or an angle greater than 90 degrees?
The calculator performs a basic sanity check: velocity must be non‑negative (zero is allowed, giving all results zero). If you enter a negative value, it will treat the absolute value after informing you, though it's best to enter positive values. Angles are taken modulo 360° and interpreted accordingly; the formulas still hold, but the physical interpretation may change.
Q5. Are air resistance and wind considered in these calculations?
No. This is an idealised projectile motion model that neglects air drag, wind, and any variation in gravity with height. It is the exact model used in JEE Main/Advanced and NCERT problems, where these effects are ignored.