The equations of motion over time

Setting up the reference frame

We choose a reference frame with the origin at the launch point, the x axis horizontal (direction of motion) and the y axis vertical, pointing upward. The projectile is launched with an initial speed v0 making an angle alpha with the horizontal.

The components of the initial speed are:

• v0x = v0 * cos(alpha)
• v0y = v0 * sin(alpha)

The equations of motion

By applying the laws of motion (constant speed on x, acceleration -g on y), we obtain at time t:

Speed:

• vx(t) = v0 * cos(alpha)
• vy(t) = v0 * sin(alpha) - g*t

Position (starting from the origin, y0 = 0):

• x(t) = v0 * cos(alpha) * t
• y(t) = v0 * sin(alpha) * t - (1/2) * g * t^2

Concrete example

A soccer player kicks a ball with v0 = 10 m/s and alpha = 30 degrees. We have cos(30) = 0.87 and sin(30) = 0.5. So v0x = 8.7 m/s and v0y = 5 m/s.

After t = 0.5 s:

• x = 8.7 * 0.5 = 4.35 m
• y = 5 * 0.5 - 0.5 * 9.8 * 0.5^2 = 2.5 - 1.225 = 1.275 m

Summary table

Quantity	x axis	y axis
Initial speed	v0*cos(alpha)	v0*sin(alpha)
Acceleration	0	-g
Position at time t	v0*cos(alpha)*t	v0*sin(alpha)*t - (1/2)gt^2

Common pitfall

Watch out for angle units (degrees vs radians depending on the calculator) and the sign of g: since the y axis points upward, the acceleration due to gravity must be written as -g, otherwise all computed heights will be wrong.