To start an avalanche on a mountain slope, an artillery shell is fired with an initial velocity of 340 m/s at 45.0° above the horizontal. It explodes on the mountainside 41.0 s after firing. What are the x and y coordinates of the shell where it explodes, relative to its firing point?
The horizontal velocity component is
Vx = 340 cos 45 = 240.4 m/s.
x coordinate = 240.4 m/s x 41.1 s = __
For the y coordinate, use the initial vertical velocity component
Vyo = 340 sin 45 = 240.4 m/s
y = Voy*41.2 - (1/2) g*(41.2)^2
jhhhkj
To find the x and y coordinates of the shell where it explodes, we can break down the initial velocity into its horizontal and vertical components.
Given:
Initial velocity (v₀) = 340 m/s
Launch angle (θ) = 45.0°
Time of flight (t) = 41.0 s
Step 1: Find the horizontal component of the initial velocity (v₀x).
v₀x = v₀ * cos(θ)
Substituting the given values:
v₀x = 340 * cos(45.0°)
v₀x = 240.45 m/s (approx.)
Step 2: Find the time the shell spends in the air horizontally (t₁).
t₁ = t = 41.0 s
Step 3: Find the horizontal displacement (x).
x = v₀x * t₁
Substituting the values:
x = 240.45 m/s * 41.0 s
x = 9869.45 m (approx.)
Therefore, the x coordinate of the shell where it explodes is approximately 9869.45 m.
Step 4: Find the vertical component of the initial velocity (v₀y).
v₀y = v₀ * sin(θ)
Substituting the given values:
v₀y = 340 * sin(45.0°)
v₀y = 240.45 m/s (approx.)
Step 5: Find the time the shell spends in the air vertically (t₂).
t₂ = t = 41.0 s
Step 6: Find the vertical displacement (y).
y = v₀y * t₂ + (1/2) * g * t₂²
Here, g represents the acceleration due to gravity, which is approximately 9.8 m/s².
Substituting the values:
y = 240.45 m/s * 41.0 s + (1/2) * 9.8 m/s² * (41.0 s)²
y = 8444.95 m (approx.)
Therefore, the y coordinate of the shell where it explodes is approximately 8444.95 m.
Hence, the shell explodes at a location approximately 9869.45 m horizontally and 8444.95 m vertically from its firing point.
To solve this problem, we can break down the initial velocity of the shell into its horizontal and vertical components.
Given:
Initial velocity (v0) = 340 m/s
Launch angle (θ) = 45.0°
Step 1: Find the horizontal and vertical components of the initial velocity.
The horizontal component (vx) is given by:
vx = v0 * cos(θ)
The vertical component (vy) is given by:
vy = v0 * sin(θ)
Step 2: Calculate the time it takes for the shell to hit the mountainside after firing.
Given:
Time of flight (t) = 41.0 s
Since the shell is fired horizontally, the time of flight is the same as the time it takes for it to hit the mountainside.
Step 3: Calculate the horizontal distance traveled by the shell.
The horizontal distance (dx) can be found using the formula:
dx = vx * t
Substitute the values of vx and t into the equation to get dx.
Step 4: Calculate the vertical distance traveled by the shell.
The vertical distance (dy) can be found using the formula:
dy = vy * t + (1/2) * g * t^2
where g is the acceleration due to gravity (approximated as 9.8 m/s^2).
Substitute the values of vy, t, and g into the equation to get dy.
Step 5: Determine the x and y coordinates of the shell where it explodes.
The x-coordinate (x) is the horizontal distance traveled and the y-coordinate (y) is the vertical distance traveled.
Therefore, the x-coordinate is dx and the y-coordinate is dy.
Calculate dx and dy using the steps outlined above to find the x and y coordinates of the explosion relative to the firing point.