ANSWER
[tex]154.6m[/tex]EXPLANATION
First, let us make a sketch of the problem:
We want to find d from the diagram above.
To do this, we have to first find the time taken for the mattress to catch the human using the formula for vertical height below:
[tex]y=y_0+v_{iy}t-\frac{1}{2}gt^2[/tex]where y = ground level = 0 m
y0 = initial height of cannon = 0.75 m
viy = initial velocity in the y-direction
t = time taken
g = acceleration due to gravity
The components of velocity for the body are shown below:
This means that the velocity in the y-direction is given as:
[tex]v_{iy}=v_i\sin \theta[/tex]where vi = initial velocity, θ = angle of launch
Therefore, we have that:
[tex]\begin{gathered} 0=0.75+40\sin 37(t)-\frac{1}{2}\cdot10\cdot t^2 \\ \Rightarrow0=0.75+24.07t-5t^2 \\ \Rightarrow5t^2-24.07t-0.75=0 \end{gathered}[/tex]Solving for t using quadratic formula, we have that:
[tex]t=4.84s;t=-0.031s[/tex]Since time has to be positive, we have that:
[tex]t=4.84s[/tex]Now, we have to find the distance of the mattress from the cannon by applying the formula:
[tex]d=v_{ix}t[/tex]where vix = initial velocity in the x-direction.
We have that:
[tex]v_{ix}=v_i\cos \theta[/tex]Therefore, the distance is:
[tex]\begin{gathered} d=v_i\cos \theta\cdot t \\ d=40\cdot\cos 37\cdot4.84 \\ d=154.6m \end{gathered}[/tex]
