Question

question 5
6 points
you are laying on the ground (y = 0) and you want to determine the firing range. if you know the nerf gun launched darts at a speed of 4 m/s and you launch the dart at 40° above the horizontal what horizontal distance (in meters) will the dart travel?

Explanation:

Step1: Resolve initial velocity components

The initial velocity $v_0 = 4$ m/s. The initial - horizontal velocity $v_{0x}=v_0\cos\theta$ and the initial - vertical velocity $v_{0y}=v_0\sin\theta$, where $\theta = 40^{\circ}$.
$v_{0x}=4\cos40^{\circ}\approx4\times0.766 = 3.064$ m/s, $v_{0y}=4\sin40^{\circ}\approx4\times0.643 = 2.572$ m/s.

Step2: Find the time - of - flight

The vertical displacement $y - y_0 = v_{0y}t-\frac{1}{2}gt^2$. Since $y_0 = 0$ and $y = 0$ (starts and ends at ground - level), we have the equation $0 = v_{0y}t-\frac{1}{2}gt^2=t(v_{0y}-\frac{1}{2}gt)$. One solution is $t = 0$ (corresponds to the launch time). The other non - zero solution is $t=\frac{2v_{0y}}{g}$.
Taking $g = 9.8$ m/s², $t=\frac{2\times2.572}{9.8}\approx0.525$ s.

Step3: Calculate the horizontal range

The horizontal range $R = v_{0x}t$. Substituting $v_{0x}=3.064$ m/s and $t = 0.525$ s, we get $R=3.064\times0.525\approx1.61$ m.

Answer:

1.61 m