Projectile Motion Calculator
Calculate trajectory parameters for objects in projectile motion
Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity and air resistance (which we neglect in ideal conditions). The path followed by a projectile is called its trajectory, which is parabolic in shape under ideal conditions.
Key characteristics of projectile motion include:
- Two-dimensional motion combining horizontal and vertical components
- Constant horizontal velocity (neglecting air resistance)
- Constant vertical acceleration due to gravity
- Parabolic trajectory
Initial Conditions
- Initial velocity (v₀): The speed at which the object is launched
- Launch angle (θ): The angle between the initial velocity vector and the horizontal
- Initial height (h₀): The height from which the object is launched
Motion Components
- Horizontal motion: Constant velocity motion
- Vertical motion: Uniformly accelerated motion due to gravity
- Combined motion: Results in a parabolic path
Initial Velocity Components
- v₀ₓ = v₀ cos(θ) [Horizontal component]
- v₀ᵧ = v₀ sin(θ) [Vertical component]
Maximum Height
h_max = h₀ + (v₀ᵧ)²/(2g)
Time of Flight
t_total = [v₀ᵧ + √(v₀ᵧ² + 2gh₀)]/g
Range
R = v₀ₓ × t_total
Example 1: Maximum Range
A ball is launched at 10 m/s at a 45° angle from ground level. The maximum range will be:
- Initial velocity components: v₀ₓ = v₀ᵧ = 7.07 m/s
- Time of flight: 1.44 seconds
- Range: 10.2 meters
Example 2: Maximum Height
For the same ball, the maximum height reached will be:
- Using h_max = (v₀ᵧ)²/(2g)
- Maximum height = 2.55 meters
What angle gives the maximum range?
For a projectile launched from ground level, 45° gives the maximum range. This is because it provides the optimal balance between horizontal and vertical velocity components.
Why do we ignore air resistance?
In basic projectile motion calculations, we often ignore air resistance to simplify the mathematics. This gives us a good approximation for many situations, especially for dense objects moving at moderate speeds.
How does initial height affect the range?
Increasing the initial height increases the total time of flight, which in turn increases the range. The relationship is not linear due to the influence of gravity.