A shell is fired from a cliff that is 30 m above a horizontal plane. The muzzle speed of the shell is 75.0 m/s and it is fired at an elevation of 35° above the horizontal.
(a) Determine the horizontal range of the shell.
(b) Determine the velocity of the shell as it strikes the ground.
To solve this problem, we need to split the initial velocity into its horizontal and vertical components.
Given:
Initial vertical velocity (Vy) = 75.0 m/s * sin(35°)
Initial horizontal velocity (Vx) = 75.0 m/s * cos(35°)
(a) To find the horizontal range, we need to find the time of flight (t) first. The time of flight can be calculated using the following formula:
t = (2 * Vy) / g
where g is the acceleration due to gravity (9.8 m/s^2).
t = (2 * 75.0 m/s * sin(35°)) / 9.8 m/s^2
t ≈ 11.65 s
Now, we can calculate the horizontal range (R) using the formula:
R = Vx * t
R = (75.0 m/s * cos(35°)) * 11.65 s
R ≈ 713.6 m
Therefore, the horizontal range of the shell is approximately 713.6 meters.
(b) To find the velocity of the shell as it strikes the ground, we can use the following formula:
V = √(Vx^2 + Vy^2)
V = √((75.0 m/s * cos(35°))^2 + (75.0 m/s * sin(35°))^2)
V ≈ 75.0 m/s
Therefore, the velocity of the shell as it strikes the ground is approximately 75.0 m/s.
To solve this problem, we can break it down into two parts:
Part 1: Horizontal Range
To find the horizontal range (R), we can use the formula:
R = (v^2 * sin(2θ)) / g
where:
v = muzzle speed of the shell = 75.0 m/s
θ = elevation angle above the horizontal = 35°
g = acceleration due to gravity = 9.8 m/s^2
Using the given values, we can calculate the horizontal range:
R = (75.0^2 * sin(2 * 35°)) / 9.8
R = (5625 * sin(70°)) / 9.8
R = (5625 * 0.9397) / 9.8
R ≈ 540.9 m
Therefore, the horizontal range of the shell is approximately 540.9 m.
Part 2: Velocity as it Strikes the Ground
To find the velocity of the shell as it strikes the ground, we can use the horizontal component of the velocity. The horizontal component (Vx) can be calculated using the formula:
Vx = v * cos(θ)
where:
v = muzzle speed of the shell = 75.0 m/s
θ = elevation angle above the horizontal = 35°
Using the given values, we can calculate the horizontal component of the velocity:
Vx = 75.0 * cos(35°)
Vx = 75.0 * 0.8192
Vx ≈ 61.44 m/s
Therefore, the velocity of the shell as it strikes the ground is approximately 61.44 m/s.
To determine the horizontal range of the shell, we need to analyze the projectile motion. We can break down the motion into horizontal and vertical components.
(a) To find the horizontal range, we need to consider the horizontal component of the velocity. The initial horizontal velocity of the shell can be calculated using trigonometry:
V_initial_horizontal = V_initial * cos(Angle)
where V_initial is the muzzle speed of the shell (75.0 m/s) and Angle is the elevation angle (35 degrees).
So, V_initial_horizontal = 75.0 * cos(35°)
Now, we can find the time it takes for the shell to hit the ground. In projectile motion, the time of flight (t) can be calculated using the vertical component of time:
t = (2 * V_initial_vertical) / g
where V_initial_vertical is the initial vertical velocity of the shell and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To find the initial vertical velocity, we can use trigonometry again:
V_initial_vertical = V_initial * sin(Angle)
Substituting the values, we get:
V_initial_vertical = 75.0 * sin(35°)
Now, we can calculate the time of flight (t):
t = (2 * V_initial_vertical) / g
Finally, we can calculate the horizontal range (R) using the formula:
R = V_initial_horizontal * t
(b) To determine the velocity of the shell as it strikes the ground, we need to find the final velocity. The final velocity in the vertical direction (V_final_vertical) can be calculated using the formula:
V_final_vertical = V_initial_vertical - g * t
where g is the acceleration due to gravity and t is the time of flight.
The final velocity in the horizontal direction (V_final_horizontal) remains constant throughout the motion.
The magnitude of the final velocity (V_final) can be found using the Pythagorean theorem:
V_final = √(V_final_horizontal^2 + V_final_vertical^2)
You can plug in the values and solve these equations to find the answers.