An artillery shell is fired with an initial velocity of 300 m/s at 65.0° above
the
horizontal. To clear an avalanche, it explodes on a mountainside 37.5 s after
firing. What are the x and y components of the shell where it explodes, relative
to its firing point?
x = __________m
y = __________m
To find the x and y components of the shell where it explodes, we can use the equations of motion.
First, we need to break down the initial velocity into its x and y components. The initial velocity (v₀) of the shell is given as 300 m/s at an angle of 65.0° above the horizontal.
The x-component of the initial velocity (v₀x) can be found using the formula: v₀x = v₀ * cos(θ)
where θ is the angle of 65.0°. Let's calculate it:
v₀x = 300 m/s * cos(65.0°)
v₀x ≈ 300 m/s * 0.4226
v₀x ≈ 126.78 m/s
Therefore, the x-component of the initial velocity is approximately 126.78 m/s.
The y-component of the initial velocity (v₀y) can be found using the formula: v₀y = v₀ * sin(θ)
v₀y = 300 m/s * sin(65.0°)
v₀y ≈ 300 m/s * 0.9063
v₀y ≈ 271.89 m/s
Therefore, the y-component of the initial velocity is approximately 271.89 m/s.
Now, let's use these components to find the x and y coordinates of where the shell explodes, relative to its firing point.
The x-coordinate (x) at any given time (t) can be found using the formula: x = v₀x * t
x = 126.78 m/s * 37.5 s
x ≈ 4756.25 m
Therefore, the x-component of where the shell explodes, relative to its firing point, is approximately 4756.25 m.
The y-coordinate (y) at any given time (t) can be found using the formula: y = v₀y * t - 0.5 * g * t²
where g is the acceleration due to gravity (approximately 9.8 m/s²).
y = 271.89 m/s * 37.5 s - 0.5 * 9.8 m/s² * (37.5 s)²
y ≈ 10199.69 m - 0.5 * 9.8 m/s² * (1406.25 s²)
y ≈ 10199.69 m - 68662.50 m
y ≈ -58462.81 m (negative value indicates below the initial launch height)
Therefore, the y-component of where the shell explodes, relative to its firing point, is approximately -58462.81 m.
In conclusion, the x-component is 4756.25 m and the y-component is -58462.81 m, where the shell explodes relative to its firing point.