Interactive physics simulator
Horizontal Projectile Motion
Observe the motion of an object launched completely horizontally from a height. Explore how horizontal velocity remains constant while vertical speed increases, mapping a beautiful parabolic trajectory.
Horizontal Projectile Simulator
Control height, speed, gravity, and object parameters. Launch horizontally to see the components trace in real-time.
Live Result
- Initial Speed (v_0)
- 15 m/s
- Platform Height (h)
- 20 m
- Horizontal Speed (v_x)
- 15 m/s
- Vertical Speed (v_y)
- 0 m/s
- Time Elapsed (t)
- 0 s
- Position (x, y)
- (0 m, 20 m)
- Current Speed (v)
- 15 m/s
- Time of Flight
- 2.02 s
- Landing Range (R)
- 30.3 m
- Gravity (g)
- 9.8 m/s²
- Impact Angle
- 52.8°
- Active Equation
- x = v_0 × t; y = h - 1/2gt²
What is Horizontal Projectile Motion?
Horizontal projectile motion describes the path of an object that is launched horizontally from a certain elevation, with no initial vertical velocity. Once projected, the only force acting upon it is gravity (ignoring air resistance).
The classic way to understand this is to break the motion down into two completely independent directions:
- Horizontal Direction (X-Axis): The object moves at a constant speed because there is no horizontal force pushing or pulling it. Its velocity vx remains exactly equal to the launch velocity v0.
- Vertical Direction (Y-Axis): The object acts like it was simply dropped from rest. Gravity accelerates it downward at g (9.8 m/s² on Earth), causing its vertical velocity vy to increase by 9.8 m/s every second.
Key Ideas of Horizontal Launch
Horizontal and vertical motions take place simultaneously but are completely independent of each other.
- Initial Vertical Velocity: The launch angle is 0°, meaning uy is always 0 m/s.
- Independence of Motion: If you drop a ball and launch another horizontally from the same height at the same time, they will hit the ground at the exact same instant, despite the second ball traveling further horizontally.
- Constant Horizontal Velocity: Assuming no air resistance, horizontal acceleration is zero.
- Uniformly Accelerated Vertical Motion: The object falls under gravity with constant downward acceleration g.
Kinematic Formulas
vx = v0
vy = g * t
x = v0 * t
y = h - ½gt2
Important derived variables:
- Time of Flight: t = √(2h / g)
- Horizontal Range: R = v0 * √(2h / g)
- Impact Velocity Magnitude: v = √(vx2 + vy2)
- Landing Angle below horizontal: θ = arctan(vy / vx)
Component Breakdown
| Motion Property | Horizontal (X) | Vertical (Y) |
|---|---|---|
| Initial Velocity | v0 | 0 m/s |
| Velocity at time t | vx = v0 (constant) | vy = gt (downward) |
| Acceleration | ax = 0 m/s² | ay = g (downward) |
| Displacement | x = v0t | y = h - ½gt² |
| Force Acting | None | Gravity (weight) |
Real-life Example
A package is dropped from a cargo plane flying horizontally at 50 m/s at an altitude of 80 m.
Initial Horizontal Speed: v0 = 50 m/s. Vertical Height: h = 80 m.
Calculating the flight time:
t = √(2 × 80 / 9.8) = √(16.3) ≈ 4.04 s.
Calculating the landing range:
R = 50 × 4.04 ≈ 202 m.
Solved Examples
A package is dropped horizontally from a rescue airplane flying at 80 m/s at an altitude of 490 m. Find the time it takes to hit the ground and the horizontal range. Use g = 9.8 m/s².
- Horizontal launch means initial vertical velocity uy = 0 m/s.
- Use vertical position equation to find time of flight: h = ½gt², which gives t = √(2h/g).
- Substitute values: t = √(2 × 490 / 9.8) = √(980 / 9.8) = √100 = 10 s.
- Calculate horizontal range using constant horizontal speed: R = v0 × t.
- R = 80 m/s × 10 s = 800 m.
Answer: Time of Flight = 10 s, Range = 800 m
A ball rolls off a horizontal desk of height 1.25 m and lands on the floor 3.0 m away from the edge of the desk. Calculate the speed at which it rolled off the table. Use g = 10 m/s².
- Given launch height h = 1.25 m, horizontal range R = 3.0 m, and gravity g = 10 m/s².
- Calculate time of flight: t = √(2h/g) = √(2 × 1.25 / 10) = √(2.5 / 10) = √0.25 = 0.5 s.
- Since R = v0 × t, find the initial velocity: v0 = R / t.
- v0 = 3.0 m / 0.5 s = 6.0 m/s.
Answer: Initial speed = 6.0 m/s
A stunt motorcycle drives horizontally off a 20 m high platform at 15 m/s. Find the speed and angle of the motorcycle just before it lands on the ground. Use g = 9.8 m/s².
- Identify initial values: v0 = vx = 15 m/s (constant), launch height h = 20 m, gravity g = 9.8 m/s².
- Calculate time of flight: t = √(2h/g) = √(2 × 20 / 9.8) ≈ 2.02 s.
- Find the vertical component of velocity at landing: vy = g × t = 9.8 × 2.02 ≈ 19.8 m/s.
- Calculate landing speed using the Pythagorean theorem: v = √(vx2 + vy2).
- v = √(152 + 19.82) = √(225 + 392) = √617 ≈ 24.8 m/s.
- Find impact angle below the horizontal: θ = arctan(vy / vx) = arctan(19.8 / 15) ≈ 52.8°.
Answer: Speed = 24.8 m/s, at 52.8° below the horizontal
Common Mistakes
- Confusing launch velocities: Assuming there is an initial vertical speed. For horizontal projectiles, initial vy is always 0.
- Thinking horizontal velocity slows down: Assuming the object slows down horizontally as it falls. Without air resistance, vx is strictly constant.
- Using general projectile equations: Trying to use u2 sin(2θ) / g for the range. That formula only applies to projectiles launched from and landing on the same horizontal plane.
- Ignoring independence: Believing that launching an object faster makes it hit the ground later. The fall time depends solely on gravity and height.
Quick Summary
- Horizontal launch angle θ = 0°.
- Horizontal speed vx is constant. Horizontal displacement: x = v0t.
- Vertical speed vy increases by gt. Vertical displacement: y = h - ½gt2.
- Time of flight: t = √(2h/g).
- Mass of the projectile does not affect its motion in a vacuum.
- The trajectory shape is a semi-parabolic arc.
Practice Questions
1. A marble rolls off a 0.8 m high table at 2.5 m/s. How far from the table's base does it land? (Use g = 9.8 m/s²)
t = √(2 × 0.8 / 9.8) ≈ 0.404 s. Range R = v0 × t = 2.5 × 0.404 ≈ 1.01 meters.
2. An arrow is shot horizontally from a bow at a height of 1.8 m with speed 60 m/s. What is its horizontal range? (Use g = 9.8 m/s²)
t = √(2 × 1.8 / 9.8) ≈ 0.606 s. Range R = 60 × 0.606 ≈ 36.4 meters.
3. If gravity on the Moon is 1.62 m/s², how does the landing time of a horizontal projectile on the Moon compare to that on Earth from the same height?
Since t = √(2h/g), lower gravity (g) increases the time of flight. The object will stay in the air much longer on the Moon.
4. Does dropping a ball vertically and throwing another horizontally at the same speed from the same height change their ground impact times?
No. Because horizontal motion is completely independent of vertical motion, both experience the exact same vertical acceleration (g) starting from zero vertical velocity. They hit the ground at the same time.
5. What is the vertical velocity of a horizontally launched projectile at the instant of launch?
It is exactly 0 m/s. The launch velocity is entirely horizontal, and vertical velocity only increases after launch due to gravity.
6. If initial horizontal speed is doubled, how does the time of flight and range change?
The time of flight remains completely unchanged because it depends only on height and gravity. The horizontal range doubles because Range = vx × Time.
FAQ
Frequently Asked Questions
What is horizontal projectile motion?
Horizontal projectile motion is a special case of projectile motion where an object is launched with an initial velocity directed entirely horizontally (angle = 0°) from a specific height, then falls along a parabolic path due to gravity.
What is the initial vertical velocity of a horizontal projectile?
The initial vertical velocity (uy) is exactly 0 m/s because the object is fired completely horizontally.
What is the formula for the time of flight?
The time of flight (t) is determined by launch height (h) and gravity (g): t = √(2h/g). It does not depend on the launch speed.
What is the formula for the horizontal range?
The horizontal range (R) is the horizontal distance traveled during flight: R = v0 × t = v0 × √(2h/g), where v0 is the initial horizontal velocity.
Does horizontal speed affect flight time?
No. Horizontal motion and vertical motion are independent. A faster horizontal launch speed increases the range but has no effect on the time of flight.
Why is the horizontal velocity constant in projectile motion?
Because we ignore air resistance, there are no forces acting horizontally on the projectile. Since horizontal force is zero, horizontal acceleration is zero, and horizontal velocity (vx) remains constant.
How does gravity affect a horizontally launched projectile?
Gravity exerts a constant downward force, accelerating the projectile vertically at g (9.8 m/s² on Earth). This causes the vertical velocity to increase linearly (vy = gt) while the object falls.
If a bullet is shot horizontally and another is dropped from the same height, which hits the ground first?
Ignoring air resistance and Earth's curvature, they will hit the ground at the exact same time. Both begin with zero vertical velocity and experience the same downward gravitational acceleration.
Does mass affect the range or flight time?
No. In an ideal vacuum model, all objects fall with the same gravitational acceleration regardless of their mass. Therefore, mass does not affect flight time or range.
What is the path shape of a horizontal projectile?
The trajectory is a semi-parabola (one half of a symmetric parabola), created because the horizontal distance increases linearly with time while the vertical distance falls quadratically with time.
How do you find the final velocity at impact?
The final velocity is the vector sum of the constant horizontal component (vx = v0) and the final vertical component (vy = gt). The impact speed is v = √(vx2 + vy2), directed at an angle θ = arctan(vy / vx) below the horizontal.
What are some real-world examples of horizontal projectile motion?
Examples include a ball rolling off a tabletop, a package dropped from an airplane in level flight, a stone thrown straight out from a cliff, or water spraying horizontally from a garden hose.