Read and reason
A velocity is more than one number
A velocity specified as 12 m/s is incomplete if the direction matters. In two-dimensional coordinates, the launch velocity can be represented by vx0 and vy0. The magnitude tells us the initial speed. An angle θ measured counterclockwise upward from +x determines the components: vx0=v0 cos θ and vy0=v0 sin θ. These relationships use right-triangle geometry and require that θ be measured from the horizontal.
The components share a common direction convention
Choose +x as right and +y as up. A launch at +30° above +x has positive vx0 and vy0. If θ is negative, the initial vertical component points downward. In the current learning studio, angles are restricted to 0° through 80°, speed and starting height are nonnegative, and gravity points downward. This limits the situations the tool models while keeping the x/y equations explicit.
Connect components to time
With negligible drag, vx(t)=vx0 is constant while vy(t)=vy0−gt changes. The vertical velocity becomes zero at an apex, t_apex=vy0/g, only if the apex occurs before landing and vy0 is positive. The horizontal position increases linearly x(t)=x0+vx0t, while y(t)=h+vy0t−(g/2)t² is a downward-opening quadratic. Their combined positions form a curved trajectory.
Evaluate the boundary of the model
Sin and cosine are geometric definitions for vector decomposition; the same initial speed yields different horizontal and vertical components at different angles. Do not confuse a horizontal launch at 0° from nonzero height with a launch from the ground where immediate contact with ground is a boundary condition. Air drag and spin are absent here. The equations cannot certify safe real launching distances or conditions.