Vertical projectile motion in one dimension

Key points:

1. Gravity: The only force acting on the object is gravity, which causes a constant downward acceleration. This acceleration due to gravity is denoted by 'g' and is approximately $9.8 \text{ m/s}^2$ near the Earth's surface.
2. Acceleration: The acceleration is always directed downwards. If we define the upward direction as positive, then the acceleration 'a' is $-g$. If we define the downward direction as positive, then 'a' is $+g$.
3. Velocity:
   • When an object is thrown upwards, its initial velocity is positive (if up is positive).
   • As it rises, its velocity decreases due to gravity until it reaches its highest point, where its velocity is momentarily zero.
   • As it falls back down, its velocity becomes negative (if up is positive) and increases in magnitude.
4. Symmetry: For an object thrown upwards and returning to the same height, the time taken to reach the highest point is equal to the time taken to fall back down from that point. The speed at which it returns to the starting height is equal to its initial upward speed.
5. Equations of Motion: The standard kinematic equations can be used, with 'a' replaced by 'g' (and its appropriate sign):
   • $v = v₀ + at$
   • $Δy = v₀t + (1/2)at²$
   • $v² = v₀² + 2aΔy$ where:
     • $v$ is the final velocity
     • $v₀$ is the initial velocity
     • $a$ is the acceleration ($-g$ or $+g$)
     • $t$ is the time
     • $Δy$ is the displacement (change in vertical position)