What is the exit velocity of the dart? What is the acceleration of the dart inside the blowgun?

Question

A blowgun dart is fired horizontally and strikes a target 20 meters away after dropping 0.75 meters. What is the exit velocity of the dart if the blowgun is 1.15 meters long, what is the acceleration of the dart inside the blowgun?

Answer 1

Let's find time of flight from the descent:

With y axis pointing upwards, Δy = -0.75m , starting velocity v_y0= 0, acceleration a=-g= -9.8 m/s²

Δy = v_y0t + (1/2) at² = (1/2) at² i.e. t =sqrt(2Δy/a) =sqrt( 2*0.75/9.8) =0.39 s

During this time it undergoes a horizontal displacement Δx=20m at a constant horizontal velocity v_x0

Δx = v_x0 t => Exit velocity, v_x0 = Δx /t = 20m /0.39s = 51 m/s [ two significant digits]

Assuming constant acceleration within the barrel, a= (v²-v₀²)/2Δx gives

a = 51²/(2*1.15) = 1130m/s²