Write down the initial velocity (Vo) and the angle theta that the projectile is launched at. For example, for a projectile launched from ground zero with an initial speed of 40 meters per second (m/s) at an angle of 60 degrees, write Vo = 40 m/s and theta = 60 degrees.
Break the initial velocity Vo into its x and y components. Recall that in the x direction (Vox) = Vo cos (theta). If theta = 60 degrees and Vo = 40 m/s then Vox = (40 m/s) cos (60 degrees). Vox = 40 m/s (.5) or Vox = 20 m/s. In the y direction Voy = Vo sin (theta). From the example, this yields Voy = (40 m/s) sin (60 degrees) or Voy = 40 m/s (.866) or Voy = 34.64 m/s.
Find the time it takes for the projectile to reach its maximum height. Recall the basic motion equation in the y direction, V2y = Voy - gt where V2y = velocity in the y direction at the top of the trajectory = 0, Voy = initial velocity in the y direction, g = acceleration due to gravity (9.8 m/s^2), and t = time in seconds. Solve the basic equation for t and plug in given values.
V2y = Voy - gt
V2y - Voy = - gt
(V2y - Voy) /-g = t
0 - 34.64 m/s/ (-9.8 m/s^2) = t
3.53 s = t
Calculate the maximum height, the value of y at this time. Recall the equation
y = yo + Voy (t) - (1/2)g(t)^2 where yo = initial displacement in the y direction = 0
y = yo + Voy (t) - (1/2)g(t)^2
y = 0 + Voy (t) - (1/2)g(t)^2
y = Voy (t) - (1/2)g(t)^2
y = 34.64 m/s(3.53s) - (1/2) (9.8)m/s^2 (3.53s)^2
y = 122.28m - 61.06 m
y = 61.22 m
Determine the total time (T) from launch to landing. With no air resistance, the projectile will spend equal time rising and falling. Once the time to reach maximum height has been calculated it is a matter of simply doubling it to find the total time. T = 2t. For example if t = 3.53 s then the total time T = 2 (t) or T = 2 (3.53s). T = 7.06s.
Define and calculate the range (R) of the trajectory. Range refers to the displacement in the x direction. To calculate R recall the basic displacement equation, R = (Vox) T. Also recall that the velocity in the x direction does not change. Plug the values for Vox and total time T into the equation and solve. For example:
R = Vox (T)
R = 20 m/s (7.06s)
R = 141.2 m