On level ground a shell is fired with an initial velocity of 54 m/s at 3.37° above the horizontal and feels no appreciable air resistance. How far from its firing point does the shell land?

Step 1: Convert the launch angle to radians:
Given launch angle, \theta = 3.37^{\circ} , convert to radians:
\theta = 3.37 \times \frac{\pi}{180} \, \text{radians}

Step 2: Calculate the horizontal and vertical components of the initial velocity:
Initial velocity, v_0 = 54 \, \text{m/s}
- Calculate horizontal component:
v_{0x} = v_0 \cdot \cos(3.37^{\circ})
- Calculate vertical component:
v_{0y} = v_0 \cdot \sin(3.37^{\circ})

Step 3: Calculate the time of flight using the vertical component:
Acceleration due to gravity, g = 9.81 \, \text{m/s}^2
- Calculate time of flight:
t_{flight} = \frac{2 \cdot v_{0y}}{g}

Step 4: Calculate the range using the horizontal component:
- Calculate range:
R = v_{0x} \cdot t_{flight}

Answer: The shell lands approximately 34.89 meters from its firing point.