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Module 3: Forces and motion›Projectile motion (3.1.3)

Projectile motion (3.1.3)

Independence of horizontal and vertical motion, parabolic trajectories, range, maximum height, and time of flight for projectiles in A-level Physics.
4 min

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A projectile is any object that is launched with an initial velocity and then moves solely under the influence of gravity in a uniform gravitational field. In the context of projectile motion, no other forces act on a projectile, such as thrust or air resistance.

A diagram illustrating projectile motion with a curved path. The horizontal axis is labeled 'Range'. An angle θ is shown at the starting point of the projectile, with an initial velocity 'u' indicated by an arrow. The vertical component of velocity is labeled 'Vy = 0 m s⁻¹'.

In projectile motion, horizontal and vertical velocity components can be considered independently:

  • The horizontal velocity of a projectile remains constant throughout its flight, as there is no horizontal acceleration (assuming air resistance is negligible).
  • The vertical velocity changes over time due to the acceleration caused by gravity.

The independent components combine to form a parabolic trajectory characteristic of projectile motion.

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For a projectile launched at an angle, it is useful to resolve its velocity into horizontal and vertical components using trigonometry.

A diagram showing a right triangle with the vertical axis labeled 'R sin θ' and the horizontal axis labeled 'R cos θ'. An angle θ is indicated at the bottom left corner, and a vector u is shown pointing diagonally upwards to the right.

For a projectile launched with an initial velocity at an angle to the horizontal:

  • the horizontal component of velocity is
  • the vertical component of velocity is .
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The initial vertical velocity of a projectile – equal to – determines the maximum height reached and the time of flight.

The initial horizontal velocity – equal to – affects the horizontal range (how far the projectile travels).

Two graphs showing the effect of changing vertical velocity and the effect of changing horizontal velocity. The top graph has a vertical axis labeled 'y' and a horizontal axis labeled 'x', with three curves in green, blue, and red representing different vertical velocities. The bottom graph also has a vertical axis labeled 'y' and a horizontal axis labeled 'x', with three curves in green, blue, and red representing different horizontal velocities.
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Question walkthrough

Resolving Velocity into Horizontal and Vertical Components

Resolve a projectile's initial velocity into horizontal and vertical components using trigonometry.

Key terms in projectile motion:

  • Time of flight : This is the total time a projectile remains in the air, from the moment it’s launched until it hits the ground. The time of flight is determined solely by the initial vertical velocity.
  • Maximum height : The maximum height is the highest point the projectile reaches, where its vertical velocity momentarily becomes zero before it starts descending. The maximum height is determined by the initial vertical velocity.
  • Range : The range is the horizontal distance the projectile travels from the launch point until it lands. The range depends on the initial horizontal velocity and the time of flight.
A diagram illustrating projectile motion with labeled components: 'u' representing initial velocity, 'usin θ' indicating the vertical component, 'ucos θ' indicating the horizontal component, 'max height, H' showing the peak height, and 'Range, R' denoting the horizontal distance traveled.
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Three common scenarios in projectile motion:

  • Vertical projection: The projectile is launched straight up, so there is no horizontal motion. Gravity directly opposes the vertical motion, slowing the projectile until it reaches its highest point, then pulling it back down.
  • Horizontal projection: The projectile is launched horizontally with no initial vertical velocity from an initial height above the ground. Its vertical motion begins only because gravity pulls it downward.
  • Projection at an angle: The most common scenario, where the initial velocity has both horizontal and vertical components. In this case, the velocity can be resolved into separate horizontal and vertical components, with gravity affecting only the vertical component.
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Useful equations for a projectile with an initial velocity , at an angle , include:

A table displaying three physics formulas: 'Time of flight' with the formula T = 2u sin(θ) / g, 'Maximum height' with the formula H = (u sin(θ))² / 2g, and 'Horizontal range' with the formula R = 2u² sin(2θ) / g. © Medify

These equations are valid only for a projectile that returns to its initial launch height upon landing.

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Question walkthrough

Time of Flight for Same-Height Projectile Launch

Calculate the total time of flight for a projectile launched and landing at the same height.

Question walkthrough

Maximum Height of a Projectile at an Angle

Calculate the maximum height reached by a projectile launched at an angle, using its vertical component of velocity.

Question walkthrough

Time of Flight from Projectile Range

Work backward from a projectile's range and launch angle to find its initial speed and total time of flight.

Learn how the vertical and horizontal components of velocity change (or don’t change) in projectile motion.

A graph showing a curved line with arrows indicating direction. Red arrows point upwards and downwards at various points along the curve, while blue arrows point horizontally to the left and right.
Do

Remember that the vertical velocity of an object decreases with time at a constant rate during projectile motion.

The vertical velocity decreases until it reaches zero at the highest point before reversing direction.

The vertical velocity then continues to decrease at a constant rate until it returns to the surface.

The horizontal component of velocity remains constant throughout the entire motion.

A graph of a curve with black points along it. There are red arrows pointing upwards and downwards, and blue arrows pointing left and right, indicating directions along the curve.
Don't

Assume the projectile maintains the same speed throughout its flight: gravity continuously affects vertical motion.

Assume that the horizontal component of velocity decreases over time during projectile motion.

Projectile motion always ignores any effects due to drag.

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Apply trigonometry correctly to resolve the horizontal and vertical components of an object’s trajectory in projectile motion.

SOH CAH TOA is a helpful mnemonic to remember the trigonometric ratios for right triangles:

  • ,
  • ,
  • .
A diagram showing a right triangle with a hypotenuse labeled 'u' in red. The vertical side is labeled 'usin θ' in blue, and the horizontal side is labeled 'ucos θ' in blue. There is a dashed line indicating the right angle.
Do

Use cos(θ) for the horizontal component and sin(θ) for the vertical component when resolving velocity, where θ is measured from the horizontal.

A diagram illustrating a right triangle with a hypotenuse labeled 'u' in red. The vertical side is labeled 'ucos θ' in blue, and the horizontal side is labeled 'usin θ' in blue. A dashed line outlines the triangle.
Don't

Mix up sine and cosine: this will lead to incorrect calculations for range, height, and time of flight.

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Question walkthrough

Time, Height and Range for an Elevated Launch

Calculate the time of flight, maximum height, and horizontal range for a projectile launched at an angle from an elevated platform.