To start an avalanche on a mountain slope, an artillery shell is fired with an initial velocity of 350 m/s at 51.0° above the horizontal. It explodes on the mountainside 40.0 s after firing. What are the x and y coordinates of the shell where it explodes, relative to its firing point?
To find the x and y coordinates of the shell where it explodes, we can break down the initial velocity into its x and y components.
Given:
Initial velocity (v) = 350 m/s
Launch angle (θ) = 51.0°
Time of flight (t) = 40.0 s
First, let's find the x-component of the velocity:
Vx = v * cosθ
Vx = 350 m/s * cos(51.0°)
Vx = 350 m/s * 0.6249
Vx ≈ 218.72 m/s
Now let's find the y-component of the velocity:
Vy = v * sinθ
Vy = 350 m/s * sin(51.0°)
Vy = 350 m/s * 0.7818
Vy ≈ 273.63 m/s
Since the shell explodes after 40.0 seconds, we can use this time to find the distance traveled in both the x and y direction.
For x-coordinate:
x = Vx * t
x = 218.72 m/s * 40.0 s
x = 8748.8 m
For y-coordinate:
y = Vy * t - 0.5 * g * t^2
Note: g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
y = 273.63 m/s * 40.0 s - 0.5 * 9.8 m/s^2 * (40.0 s)^2
y = 10945.2 m - 7840.0 m
y ≈ 3105.2 m
Therefore, the x and y coordinates of the shell where it explodes are approximately (8748.8 m, 3105.2 m) relative to its firing point.
To find the x and y coordinates of the shell where it explodes, we need to break down the initial velocity into its horizontal and vertical components.
Step 1: Find the horizontal component of the initial velocity.
The horizontal component of the velocity can be found using the equation:
Vx = V * cos(θ)
where V is the initial velocity and θ is the angle of the projectile with respect to the horizontal.
Vx = 350 m/s * cos(51°)
Vx ≈ 227.91 m/s
Step 2: Find the time it takes for the shell to explode.
The time is given as 40.0 seconds.
Step 3: Find the vertical component of the initial velocity.
The vertical component of the velocity can be found using the equation:
Vy = V * sin(θ)
where Vy is the vertical component of the velocity.
Vy = 350 m/s * sin(51°)
Vy ≈ 267.39 m/s
Step 4: Calculate the position of the shell where it explodes.
To calculate the x-coordinate (horizontal distance), we multiply the horizontal component of the velocity by the time taken:
x = Vx * t
x = 227.91 m/s * 40.0 s
x ≈ 9116.4 m
To calculate the y-coordinate (vertical distance), we use the equation for motion with constant acceleration:
y = 0 + Vy * t + 0.5 * g * t^2
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
y = 0 + 267.39 m/s * 40.0 s + 0.5 * 9.8 m/s^2 * (40.0 s)^2
y ≈ 53878.4 m
Therefore, the x-coordinate of the shell when it explodes relative to its firing point is approximately 9116.4 m, and the y-coordinate is approximately 53878.4 m.