Finding initial velocity without time

Question

You are playing a game of angry birds. The bird is launched exactly horizontally and travels a total of 38.7 m before hitting the ground. The height at which the bird was launched is 11m. Find the bird's initial velocity.

Answer 1

First, we are given the following:

g = -9.81 m/s^2, x_0 = 0 m, x_f = 38.7 m

y_0 = 11 m, y_f = 0 m, v_0y = 0 m/s

y_f = y_0 + v_0y * t + (g * t^2 / 2), and

x_f = x_0 + v_0x * t, since v_0x = v_fx = constant,

since there is no acceleration in the x direction.

Next, we need to find t in order to then find v_0x:

y_f = y_0 + v_0y * t + (g * t^2 / 2) = y_0 + (0 m/s) * t + (g * t^2 / 2) = y_0 + (g * t^2 / 2)

y_f - y_0 = g * t^2 / 2, so t^2 = 2 * (y_f - y_0) / g, so t = √[2 * (y_f - y_0) / g],

so now lets solve the second equation for v_0x:

x_f = x_0 + v_0x * t = (0 m) + v_0x * t = v_0x * t, so v_0x = x_f / t,

we can now plug in t to get the following:

v_0x = x_f / √[2 * (y_f - y_0) / g] = x_f * √[g / 2 * (y_f - y_0)]

Finally, we can plug in the numbers to get the following:

v_0x = x_f * √[g / 2 * (y_f - y_0)] = (38.7 m) * √[(-9.8 m/s) / 2 * (0 m - 11 m)]

v_0x = (38.7 m) * √[(-9.8 m/s) / 2 * (-11 m)] = (38.7 m) * √[9.81/s / 22] ≈ 25.84 m/s.