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 can split the initial velocity of the shell into its horizontal and vertical components.
(a) The horizontal component of the initial velocity can be found using the formula:
v_x = v * cos(theta)
where v_x is the horizontal component, v is the muzzle speed, and theta is the elevation angle. Plugging in the given values, we have:
v_x = 75.0 m/s * cos(35°) = 61.22 m/s
Now, we can calculate the time it takes for the shell to hit the ground. The vertical component of the initial velocity can be found using the formula:
v_y = v * sin(theta)
where v_y is the vertical component. Plugging in the given values, we have:
v_y = 75.0 m/s * sin(35°) = 42.58 m/s
Using the formula for free fall motion, with initial velocity v_y and initial height h, we can find the time it takes for the shell to hit the ground:
h = v_y * t - (1/2) * g * t^2
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Rearranging the equation, we get:
0 = -4.9 t^2 + 42.58 t - 30
Solving this quadratic equation, we find that t ≈ 1.78 s.
Now, we can calculate the horizontal range by multiplying the horizontal component of the initial velocity by the time of flight:
range = v_x * t = 61.22 m/s * 1.78 s ≈ 109.02 m
Therefore, the horizontal range of the shell is approximately 109.02 m.
(b) To determine the velocity of the shell as it strikes the ground, we can use the equation for final velocity in free fall motion:
v_f = v_y - g * t
where v_f is the final velocity of the shell. Plugging in the given values, we have:
v_f = 42.58 m/s - 9.8 m/s^2 * 1.78 s
v_f ≈ 25.88 m/s
Therefore, the velocity of the shell as it strikes the ground is approximately 25.88 m/s.
To solve this problem, we can use projectile motion equations.
Step 1: Determine the initial vertical and horizontal components of velocity.
The initial velocity can be broken down into horizontal and vertical components using trigonometry. The horizontal component (Vx) is given by V_initial * cos(theta), and the vertical component (Vy) is given by V_initial * sin(theta).
V_initial = 75.0 m/s (given)
theta = 35 degrees (given)
Vx = V_initial * cos(theta) = 75.0 * cos(35) ≈ 61.3 m/s
Vy = V_initial * sin(theta) = 75.0 * sin(35) ≈ 43.1 m/s
Step 2: Determine the time of flight.
The time of flight (t) is the total time it takes for the shell to reach the ground. We can find it using the vertical motion equation:
Δy = Vy * t - (1/2) * g * t^2, where Δy is the vertical displacement and g is the acceleration due to gravity (approximately 9.8 m/s^2).
We know that the initial vertical displacement is -30 m (negative because it is above the ground), and we want to find the time it takes for the shell to reach the ground. So, we have:
-30 = 43.1 * t - (1/2) * 9.8 * t^2
Rearranging the equation, we get:
4.9 * t^2 - 43.1 * t - 30 = 0
Solving this quadratic equation, we get two possible solutions for t. We can discard the negative solution because time cannot be negative:
t = 4.55 s (approximately)
Step 3: Determine the horizontal range.
The horizontal range (R) is the distance traveled by the shell in the horizontal direction. We can find it using the horizontal motion equation:
R = Vx * t
Substituting the values we found, we get:
R = 61.3 * 4.55 ≈ 279.1 m
So, the horizontal range of the shell is approximately 279.1 meters.
Step 4: Determine the velocity of the shell as it strikes the ground.
The final velocity (V_final) of the shell is given by:
V_final = √(Vx^2 + Vy^2)
Substituting the values we found, we get:
V_final = √(61.3^2 + 43.1^2) ≈ 75.3 m/s
So, the velocity of the shell as it strikes the ground is approximately 75.3 meters per second.
To solve this problem, we can break it down into horizontal and vertical components using the projectile motion equations. Let's start by finding the horizontal range of the shell.
(a) Determine the horizontal range of the shell:
The horizontal range is the horizontal distance traveled by the shell before it hits the ground. We can use the equation:
Range = (initial horizontal velocity) * (time of flight)
To find the initial horizontal velocity, we need to determine the horizontal component of the initial velocity. We can do this by multiplying the muzzle speed by the cosine of the angle of elevation:
Initial horizontal velocity = muzzle speed * cos(angle of elevation)
Given that the muzzle speed is 75.0 m/s and the angle of elevation is 35 degrees, we have:
Initial horizontal velocity = 75.0 m/s * cos(35 degrees)
Now, we need to find the time of flight, which is the time it takes for the shell to hit the ground. We can use the equation:
Time of flight = (2 * initial vertical velocity) / (acceleration due to gravity)
To find the initial vertical velocity, we need to determine the vertical component of the initial velocity. We can do this by multiplying the muzzle speed by the sine of the angle of elevation:
Initial vertical velocity = muzzle speed * sin(angle of elevation)
Given that the muzzle speed is 75.0 m/s and the angle of elevation is 35 degrees, we have:
Initial vertical velocity = 75.0 m/s * sin(35 degrees)
Now, we can substitute the values into the equation for time of flight:
Time of flight = (2 * initial vertical velocity) / (acceleration due to gravity)
where the acceleration due to gravity is approximately 9.8 m/s².
Finally, we can substitute the values into the equation for range:
Range = (initial horizontal velocity) * (time of flight)
After calculating these values, you will get the horizontal range of the shell.
(b) Determine the velocity of the shell as it strikes the ground:
To find the velocity of the shell as it strikes the ground, we can use the equation:
Final velocity = square root((initial horizontal velocity)² + (initial vertical velocity - (acceleration due to gravity * time of flight))²)
where the initial horizontal velocity, initial vertical velocity, acceleration due to gravity, and time of flight are already calculated as mentioned above.
After substituting these values into the equation, you will find the velocity of the shell as it strikes the ground.
Remember to use appropriate units and take into account significant figures throughout the calculations.