Projectile motion and wave basics: how trajectories and wave speed work

A projectile is two simple motions in one

Once a projectile leaves the hand, bat or nozzle, gravity is the only force the ideal model keeps. That force acts vertically, so it cannot change the horizontal velocity at all. The launch velocity is therefore split into a horizontal component, vx = v₀ × cos θ, and a vertical component, vy = v₀ × sin θ, where θ is the angle above the horizontal.

The two components then evolve independently. Horizontally, position is x = vx × t, a straight-line constant-velocity motion. Vertically, position is y = h₀ + vy × t − ½ × g × t², where h₀ is the release height and g is the gravitational acceleration. Combining those two equations gives the familiar parabola, but each half can be solved on its own.

A worked example makes the split concrete. Launch a ball at 20 m/s and 45 degrees from ground level: vx and vy are both about 14.14 m/s, because cos 45° and sin 45° are equal. The horizontal part covers equal distances every second; the vertical part loses about 9.81 m/s of upward speed every second until it reverses.

Time of flight, apex and range

Time of flight is the time until the projectile returns to the ground, y = 0. From that condition, t = (vy + √(vy² + 2g·h₀)) ÷ g. When h₀ is zero the formula reduces to 2vy ÷ g, the familiar up-and-down time; a launch from a height stays in the air longer because the extra fall is added under the square root.

The highest point is reached when the vertical velocity becomes zero, at t = vy ÷ g. Its height above the ground is h₀ + vy² ÷ (2g). For the 20 m/s, 45-degree example launched from the ground, the apex is about 10.2 m at 1.44 s, and the full flight lasts about 2.88 s.

Range is horizontal speed multiplied by flight time: range = vx × t. For the same example that is about 40.8 m. On level ground without air resistance, 45 degrees gives the maximum range, and two angles that add to 90 degrees, such as 30 and 60 degrees, land at the same distance with different peak heights and flight times.

What the ideal model leaves out

The equations above assume no air resistance, a flat landing surface at a constant height, constant gravity and no spin or wind. Those assumptions are excellent for short, heavy throws at modest speed and increasingly poor for light objects, high speeds or long flights. Drag grows quickly with speed and depends on the shape and surface of the object, so a real ball falls shorter than the ideal range.

Drag also breaks the symmetry of the parabola: the descent is steeper than the climb, the apex arrives slightly earlier than the symmetric model predicts, and the impact angle is steeper than the launch angle. Wind adds a horizontal force that can help or hurt the range, and a spinning ball can curve sideways through the Magnus effect, which is outside a two-dimensional model entirely.

None of that makes the calculation useless. An ideal result is a clear first estimate and a way to understand which variable dominates: doubling the launch speed roughly quadruples the range in the ideal model, while increasing the launch height lengthens the flight without changing the horizontal speed. Treat the number as a starting point, then refine it with measurement or a numerical simulation that includes drag.

Waves: speed, frequency and period

A wave carries a disturbance through a medium. Its speed is how fast that disturbance travels, its frequency is how many complete cycles pass a point per second, and its wavelength is the distance between corresponding points on consecutive cycles. The three are linked by v = f × λ, with frequency in hertz, wavelength in meters and speed in meters per second.

The period is the time for one complete cycle, T = 1 ÷ f. A 250 Hz wave completes a cycle every 0.004 seconds, or 4 ms. A 440 Hz musical note has a period of about 2.27 ms. Period and frequency are reciprocals, so doubling the frequency halves the period; this is why an oscilloscope's time-per-division setting and the signal's frequency are so tightly connected.

Sound in air near room temperature travels at roughly 343 m/s, so a 440 Hz tone has a wavelength of about 0.78 m: rearrange v = f × λ to λ = v ÷ f and divide 343 by 440. Light in a vacuum travels at about 3.0 × 10⁸ m/s, which is why a visible wavelength of 600 nm corresponds to a frequency near 5.0 × 10¹⁴ Hz.

The medium decides the speed

For a wave, the speed is a property of the medium rather than a free choice. Sound travels faster in water and much faster in steel than in air, and light travels slower in glass than in a vacuum. Frequency, by contrast, is set by the source. When a wave crosses into a new medium, its frequency usually stays the same while its wavelength changes to match the new speed, which is the basis of refraction.

That is why a wave-speed calculator needs the speed as an input in most practical problems: it cannot be derived from frequency and wavelength alone, because those two are what adjust around whatever speed the medium imposes.

A useful habit for both subjects is to write the known values with units before touching a formula, convert to SI units, and then ask whether the answer is plausible. A projectile range of 4,000 m from a hand throw, or a wave speed of 3 m/s for a radio signal, is a signal that a unit or a formula was applied in the wrong place.

• Resolve the launch velocity into components before using kinematics.
• Convert to SI units before applying v = f × λ or T = 1 ÷ f.
• State the assumptions: no drag, flat ground, constant gravity, steady wave.
• Check magnitudes against experience rather than accepting a number blindly.