Motion in two dimensions describes objects that move both horizontally and vertically at the same time, such as a kicked soccer ball, a thrown stone, or a satellite passing overhead. Instead of using one position number, we describe the motion with x and y coordinates and vectors. This matters because many real motions cannot be understood by looking in only one direction.
Breaking the motion into components makes complicated paths easier to analyze.
Key Facts
- Position in two dimensions can be written as r = x i + y j, where i and j are unit vectors in the horizontal and vertical directions.
- Velocity is the rate of change of position: v = Δr/Δt, with components vx = Δx/Δt and vy = Δy/Δt.
- For projectile motion without air resistance, horizontal velocity is constant: ax = 0 and vx = v0x.
- Vertical motion has constant acceleration due to gravity: ay = -g, where g = 9.8 m/s^2 near Earth's surface.
- Projectile component equations are x = x0 + v0x t and y = y0 + v0y t - 1/2 g t^2.
- Initial velocity components are v0x = v0 cos θ and v0y = v0 sin θ when θ is measured above the horizontal.
Vocabulary
- Vector
- A quantity with both magnitude and direction, such as displacement, velocity, or acceleration.
- Component
- One part of a vector along a chosen axis, usually the horizontal x direction or vertical y direction.
- Projectile motion
- The curved motion of an object launched into the air when gravity is the only significant force after launch.
- Trajectory
- The path followed by a moving object through space.
- Acceleration due to gravity
- The downward acceleration of a freely falling object near Earth, approximately 9.8 m/s^2.
Common Mistakes to Avoid
- Using the total velocity in one-dimensional equations, which is wrong because horizontal and vertical motions must be handled with separate components.
- Assuming horizontal velocity changes during ideal projectile motion, which is wrong because gravity acts vertically and ax = 0 when air resistance is ignored.
- Forgetting the negative sign for gravity, which is wrong when upward is chosen as positive because the acceleration points downward.
- Using sin and cos on the wrong components, which is wrong because for an angle measured above the horizontal, v0x = v0 cos θ and v0y = v0 sin θ.
Practice Questions
- 1 A ball is launched at 20 m/s at 30 degrees above the horizontal. Find its initial horizontal and vertical velocity components.
- 2 A projectile is fired horizontally from a 45 m high cliff with an initial speed of 12 m/s. Ignoring air resistance, find the time to hit the ground and the horizontal distance traveled.
- 3 Two balls are released from the same height at the same time. One is dropped straight down and the other is launched horizontally. Explain which ball hits the ground first and why, assuming no air resistance.