Motion in Two Dimensions

Projectile motion and planar trajectories combining independent orthogonal components.

General idea of the topic

In two dimensions the motion is decomposed into two independent coordinates, linked only by time. The fundamental case is the projectile: without air resistance, the horizontal component is uniform and the vertical one is uniformly accelerated with the acceleration of gravity.

The topic first treats horizontal projection, which starts with only a horizontal velocity from a certain height, and then oblique projection, with a launch angle, in which range, maximum height and flight time appear.

What you will truly understand

By the end you will be able to resolve the initial velocity, write the equations of each coordinate, eliminate time to obtain the parabolic trajectory and compute range, maximum height and impact speed.

You will also know why the maximum range on level ground is achieved at 45°.

Recommended learning route

1. Horizontal Projectile Motion: horizontal launch from a height, fall time independent of the horizontal velocity and range.
2. Projectile Motion: launch at an angle, resolution of the velocity, maximum height, flight time and range.

Physical criterion you must acquire

The essential criterion is to solve each axis separately and join them with time. On the horizontal axis there is no acceleration and on the vertical it is that of gravity, directed downward. The velocity at the highest point is not zero: only its vertical component vanishes. The result is checked by symmetry: at equal height, the arrival speed equals the departure speed.

Connection with real problems

• ball, javelin and shot put.
• ballistics and projectiles.
• water jets and fountains.
• jumps and falls.

Final goal of the topic

The final goal is to solve projectile problems with confidence, compose independent motions and obtain trajectory, range and times.