Given:
Velocity = 40 m/s
Angle = 37°
Let's find the distance from the cnnon where you should place a matress to catch the human if the muzzle is 0.75 meters from the ground.
Let's first draw the projectile motion diagram:
To find the distance, first apply the formula to solve for the time taken for the human to reach the matress.
[tex]y=y_o+v_{oy}t-\frac{1}{2}gt^2[/tex]Where:
Height from the ground to the muzzle = yo = 0.75
y = 0
Velocity = vo = 40 m/s
initial vertical velocity = voy = vo sinθ = 40 sin37
We have:
[tex]v_{oy}=v_o\sin \theta=40\sin 37=24.07\text{ m/s}[/tex]g is the acceleration due to gravity = 9.8 m/s²
Now, let's solve for the time t:
[tex]\begin{gathered} y=y_o+v_{oy}t-\frac{1}{2}gt^2 \\ \\ 0=0.75+24.07t-\frac{1}{2}\times9.8\times t^2 \\ \\ t=4.94\text{ seconds} \end{gathered}[/tex]To sove for the horizontal distance, x, apply the formula:
[tex]\begin{gathered} x=v_{ox}t \\ \\ \text{Where:} \\ v_{ox}=v_o\cos \theta=40\cos 37=31.94\text{ m/s} \end{gathered}[/tex]Hence, we have:
[tex]\begin{gathered} x=31.94\times4.94 \\ \\ x=157.8\text{ m} \end{gathered}[/tex]Therefore, the matress should be placed at 157.8 meters from the cannon in order to catch the human.
ANSWER:
157.8 meters
