(a) independence of the vertical and horizontal motion of a projectile

The idea

A projectile is any object moving freely under gravity alone, and its motion splits into two perpendicular components that do not affect each other. Horizontally there is no resultant force, so horizontal velocity stays constant and horizontal displacement is velocity × time. Vertically the only force is weight, giving constant downward acceleration g, so vertical velocity changes steadily and vertical displacement follows the constant-acceleration equations. Because the two directions are independent, the time of flight is fixed entirely by the vertical motion, while the horizontal range is fixed by the horizontal velocity and that same time.

For example: A ball rolled off a bench and one dropped vertically from the same height reach the floor together, because both have identical vertical motion.

What earns the marks

• Horizontal and vertical components of a projectile's motion are independent of each other.
• Horizontally there is no resultant force, so horizontal velocity is constant and horizontal displacement = horizontal velocity × time.
• Vertically the only force is weight, giving constant downward acceleration of magnitude g, so vertical velocity changes at a steady rate.
• The time of flight is determined solely by the vertical motion; the horizontal range follows from horizontal velocity and that time.
• A projectile released horizontally and one dropped from the same height land at the same time because their vertical motions are identical.

Common mistakes and how to put them right

• Believing that a larger horizontal velocity makes a projectile fall more slowly.

  Put it right: The horizontal and vertical motions are independent, so the vertical fall time is unchanged.

• Treating horizontal velocity as decreasing during flight.

  Put it right: With no horizontal resultant force it stays constant, and only the vertical velocity changes.

• Assuming the vertical acceleration changes direction or magnitude during flight.

  Put it right: Near the Earth's surface it stays constant at g downwards throughout.

• Thinking the two components must be combined before the time of flight can be found.

  Put it right: The vertical motion alone gives the time of flight.

Exam tips

• State explicitly that the two perpendicular components are independent before using any equation.
• Choose the vertical direction to find time of flight, then use that time with constant horizontal velocity to find range.
• Check the direction convention for g and keep signs consistent when substituting into constant-acceleration equations.