Interactive physics simulator
Projectile Motion
Projectile motion is two-dimensional motion where an object moves horizontally while gravity accelerates it downward. Horizontal velocity stays constant, while vertical velocity changes with time.
Projectile Motion Simulator
Change launch speed, angle, height, and gravity to see the trajectory, range, maximum height, and time of flight update live.
Live Result
- Initial Speed
- 22 m/s
- Angle
- 42°
- Horizontal Velocity
- 16.4 m/s
- Vertical Velocity
- 14.7 m/s
- Timer
- 0 s
- Position
- (0 m, 0 m)
- Current Speed
- 22 m/s
- Time of Flight
- 3 s
- Maximum Height
- 11 m
- Range
- 49 m
- Gravity
- 9.8 m/s²
- Formula
- x = u cos θ × t; y = h + u sin θ × t - 1/2gt²
What is Projectile Motion?
Projectile motion is motion after an object is launched and gravity is the main force acting on it. In the ideal school-level model, air resistance is ignored. The motion can be studied by separating it into two independent parts: horizontal motion and vertical motion.
The horizontal velocity stays constant because there is no horizontal acceleration. The vertical velocity changes because gravity pulls the projectile downward with acceleration g ≈ 9.8 m/s2 near Earth.
Key Idea
Projectile motion is two-dimensional motion caused by a launch velocity and downward gravitational acceleration.
- Horizontal motion has constant velocity when air resistance is ignored.
- Vertical motion has constant downward acceleration due to gravity.
- The path is usually a parabola.
- At maximum height, vertical velocity is zero for an instant.
- The projectile's mass does not change the ideal path.
Projectile Motion Formulas
vx = u cos θ
vy = u sin θ
x = vxt
y = h + vyt - 1/2gt2
For launch and landing at the same height:
- Time of flight: T = 2u sin θ / g
- Maximum height: H = u2sin2θ / 2g
- Range: R = u2sin 2θ / g
Horizontal vs Vertical Motion
| Feature | Horizontal | Vertical |
|---|---|---|
| Velocity | Constant vx | Changes with time |
| Acceleration | 0 m/s2 | -g downward |
| Formula | x = vxt | y = h + vyt - 1/2gt2 |
| Meaning | How far forward | How high above ground |
Real-life Example
A basketball shot, a kicked football, a water fountain stream, and a launched science-lab ball all follow projectile motion when gravity is the main force.
If a ball is launched at 20 m/s at 30°, its horizontal and vertical velocity components are calculated separately.
Solved Examples
A ball is launched at 20 m/s at an angle of 30° from level ground. Find the time of flight, range, and maximum height. Use g = 9.8 m/s².
- Resolve the velocity: ux = u × cos(θ) = 20 × cos(30°) = 17.3 m/s, and uy = u × sin(θ) = 20 × sin(30°) = 10 m/s.
- Time of flight for same launch and landing height: T = 2uy / g = 2 × 10 / 9.8 = 2.04 s.
- Range: R = ux × T = 17.3 × 2.04 = 35.3 m.
- Maximum height: H = uy2 / 2g = 102 / (2 × 9.8) = 5.1 m.
Answer: T = 2.04 s, R = 35.3 m, H = 5.1 m
A projectile is fired at 25 m/s at 45° from level ground. Find its range.
- For level ground, R = u2 × sin(2θ) / g.
- Substitute values: R = 252 × sin(90°) / 9.8.
- R = 625 / 9.8 = 63.8 m.
Answer: Range = 63.8 m
A ball is launched at 30 m/s at 60°. Find the horizontal and vertical components of initial velocity.
- Horizontal component: ux = u × cos(θ) = 30 × cos(60°) = 15 m/s.
- Vertical component: uy = u × sin(θ) = 30 × sin(60°) = 26.0 m/s.
- The horizontal component stays constant if air resistance is ignored.
Answer: u<sub>x</sub> = 15 m/s, u<sub>y</sub> = 26.0 m/s
A stone is thrown horizontally from a 20 m high cliff at 12 m/s. Find the time to hit the ground and horizontal range.
- Horizontal launch means uy = 0 m/s.
- Use vertical motion: 0 = h - ½gt2, so t = √(2h/g).
- t = √(2 × 20 / 9.8) = 2.02 s.
- Horizontal range: R = ux × t = 12 × 2.02 = 24.2 m.
Answer: Time = 2.02 s, Range = 24.2 m
Common Mistakes
- Using total launch speed as horizontal speed instead of calculating u cos θ.
- Forgetting that vertical velocity changes because of gravity.
- Using range formula R = u2sin 2θ / g when launch and landing heights are not the same.
- Thinking horizontal velocity becomes zero at the top of the path.
- Forgetting units: speed in m/s, time in seconds, height and range in meters.
- Confusing displacement with total curved path length.
Quick Summary
- Projectile motion combines horizontal and vertical motion.
- Horizontal velocity stays constant without air resistance.
- Vertical velocity changes due to gravity.
- The path is parabolic in the ideal model.
- Range is horizontal distance traveled before landing.
- Maximum height occurs when vertical velocity becomes zero.
- For level-ground launches, 45° gives maximum range.
Practice Questions
1. A ball is launched at 10 m/s at 30°. What are u<sub>x</sub> and u<sub>y</sub>?
ux = 10 × cos(30°) = 8.66 m/s. uy = 10 × sin(30°) = 5 m/s.
2. A projectile has u<sub>y</sub> = 14 m/s and lands at the same height. Find time of flight on Earth.
T = 2uy / g = 2 × 14 / 9.8 = 2.86 s.
3. A projectile is launched at 20 m/s at 45° on level ground. Which angle gives the same range besides 45°?
For level ground, complementary angles give equal range. Since 45° complements itself, 45° is the unique maximum-range angle.
4. If launch speed stays the same, what happens to range when gravity increases?
Range decreases because R = u2 × sin(2θ) / g for level ground.
5. Does horizontal velocity change during ideal projectile motion?
No. If air resistance is ignored, horizontal velocity remains constant.
6. What causes the curved path of a projectile?
Gravity accelerates the projectile downward while horizontal motion continues, creating a curved parabolic path.
FAQ
Frequently Asked Questions
What is projectile motion?
Projectile motion is two-dimensional motion where an object moves horizontally while gravity accelerates it downward. Its path is usually a parabola when air resistance is ignored.
What is the formula for projectile motion?
The main formulas are x = u cos θ × t and y = h + u sin θ × t - 1/2gt². These describe horizontal and vertical position at time t.
Why is projectile motion curved?
The path is curved because horizontal velocity continues while gravity changes the vertical velocity every second.
Does horizontal velocity change in projectile motion?
In ideal projectile motion without air resistance, horizontal velocity is constant because no horizontal acceleration is acting.
Does vertical velocity change in projectile motion?
Yes. Vertical velocity changes because gravity accelerates the projectile downward.
What is time of flight?
Time of flight is the total time the projectile stays in the air before it reaches the ground or target height.
What is range in projectile motion?
Range is the horizontal distance traveled by the projectile before it lands.
What is maximum height?
Maximum height is the highest vertical point reached by the projectile. At this point, vertical velocity is zero for an instant.
What angle gives maximum range on level ground?
When launch and landing heights are the same and air resistance is ignored, 45° gives the maximum range.
Does mass affect projectile motion?
In ideal projectile motion without air resistance, mass does not affect the path. Gravity gives the same acceleration to all objects.
How does the projectile simulator calculate the path?
The simulator separates motion into horizontal and vertical parts. It calculates x with horizontal velocity and y with vertical velocity plus gravity.
Can projectile motion start from a height?
Yes. If launch height is above ground, the time of flight and range are larger than a same-speed launch from ground level.