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Answer :
Final answer:
The x-coordinate of the shell where it explodes, relative to its firing point, is 10134.4 meters. The y-coordinate of the shell where it explodes is 0 meters. The x-component of the velocity just before it hits the mountainside is 183.152 m/s. The y-component of the velocity just before it hits the mountainside is -269.922 m/s.
Explanation:
To find the x and y coordinates of the shell where it explodes, we need to analyze the horizontal and vertical components of its motion.
Let's start with the horizontal motion. The initial velocity of the shell can be broken down into its x and y components using trigonometry. The x-component of the velocity can be calculated as:
Vx = V * cos(theta)
where V is the initial velocity of the shell and theta is the angle of elevation.
Substituting the given values:
Vx = 328 m/s * cos(57.1 degrees)
Vx = 328 m/s * 0.559
Vx = 183.152 m/s
Now, let's move on to the vertical motion. The y-component of the velocity can be calculated as:
Vy = V * sin(theta)
Vy = 328 m/s * sin(57.1 degrees)
Vy = 328 m/s * 0.829
Vy = 271.112 m/s
Since the shell explodes on a mountainside, its y-coordinate at the point of explosion will be zero. We can use the equation of motion to find the time it takes for the shell to reach this point:
y = Vyt - (1/2)gt^2
where y is the vertical displacement, Vy is the vertical component of the velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
Substituting the given values:
0 = (271.112 m/s)t - (1/2)(9.8 m/s^2)t^2
0 = 271.112t - 4.9t^2
4.9t^2 - 271.112t = 0
t(4.9t - 271.112) = 0
t = 0 (initial time) or t = 55.33 seconds
Since the shell explodes 42.1 seconds after firing, we can conclude that it hits the mountainside at t = 55.33 seconds.
Now, let's find the x-coordinate of the shell at t = 55.33 seconds. The x-coordinate can be calculated using the equation:
x = Vx * t
x = 183.152 m/s * 55.33 s
x = 10134.4 m
Therefore, the x-coordinate of the shell where it explodes is 10134.4 meters.
Finally, to find the y-component of the velocity just before the shell hits the mountainside, we can use the equation:
Vy = Vy - gt
Vy = 271.112 m/s - 9.8 m/s^2 * 55.33 s
Vy = 271.112 m/s - 541.034 m/s
Vy = -269.922 m/s
Therefore, the y-component of the velocity just before the shell hits the mountainside is -269.922 m/s.
Learn more about projectile motion here:
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Rewritten by : Brahmana