Projectile motion (two-dimensional kinematics)

College Physics I. Also known as: phys 101, phys 1401, phys 2425, phys 1, college physics, college physics i, algebra-based physics, physics 1, general physics 1, general physics i, introductory physics i, phys 201

Splitting motion into horizontal and vertical parts to find range, height, time of flight and landing velocity.

Two independent motions

A projectile is an object moving through the air under gravity alone. The key idea is that horizontal and vertical motion are independent. Horizontally there is no force once the object has been launched, so the horizontal velocity stays constant. Vertically gravity gives a constant downward acceleration of 9.8 m/s². We ignore air resistance throughout this course unit unless a problem says otherwise.

Setting up a problem

Pick axes: x horizontal, y upward. Split the launch velocity v₀ at angle θ above the horizontal into components:

  • v₀x = v₀ cos θ (constant for the whole flight)
  • v₀y = v₀ sin θ (changes by −9.8 m/s every second)

Then treat each direction with the one-dimensional equations. Time is the link between them: the same time t is used for x and y. First find t from the vertical motion, then use it in x = v₀x t.

Useful results for level ground

When the landing height equals the launch height:

  • Time of flight: T = 2 v₀ sin θ / g
  • Maximum height: H = (v₀ sin θ)² / (2g)
  • Range: R = v₀² sin 2θ / g

The range is largest at 45 degrees, and angles that add to 90 degrees (such as 30 and 60) give the same range. At the top of the path the vertical velocity is zero but the horizontal velocity is not, so the speed there is the smallest during the flight.

Horizontal launches

An object launched horizontally has v₀y = 0. It falls through a height h in the time t = √(2h / g), the same time as an object simply dropped from that height. So a ball rolled off a table and a ball dropped from the table's edge hit the floor together. Its landing speed combines both components: speed = √(vₓ² + v_y²).

Worked example

A ball is kicked at 15 m/s at 35 degrees above the horizontal from level ground.

  1. Components: vₓ = 15 cos 35 = 12.29 m/s, v_y = 15 sin 35 = 8.60 m/s.
  2. Time of flight: T = 2 × 8.60 / 9.8 = 1.76 s.
  3. Range: R = 12.29 × 1.76 = 21.6 m.
  4. Maximum height: H = 8.60² / (2 × 9.8) = 3.78 m.

Common mistakes

  • Adding gravity into the horizontal motion.
  • Believing the speed is zero at the top of the path.
  • Using the launch speed instead of its vertical component in the height formula.
  • Using the range formula when the landing height differs from the launch height.

Practice questions

  1. A marble rolls off a table 1.2 m high at a horizontal speed of 3.0 m/s. Ignoring air resistance (g = 9.8 m/s²), how far from the table's edge does it land?

  2. A ball is launched from level ground at 20 m/s and 30 degrees above the horizontal. Ignoring air resistance (g = 9.8 m/s²), what is its horizontal range?

  3. Ignoring air resistance, a projectile is launched from level ground at the same speed each time. Which launch angle gives the greatest range?

  4. A ball flies through the air with no air resistance after being hit. Which quantity stays constant during the flight?

  5. A ball is kicked from level ground at 25 m/s at 40 degrees above the horizontal. Ignoring air resistance (g = 9.8 m/s²), what maximum height does it reach?

  6. A ball is thrown from level ground at 18 m/s at 50 degrees above the horizontal. Ignoring air resistance (g = 9.8 m/s²), how long until it lands?