Ex.4: A medieval prince locked in a castle attaches a note to a rock and throws it
at 12 m/s [25° above the horizon] from atop the castle's outer wall at a height
of 9.5 m so it just clears the surrounding moat. Determine: a) the rock's time of
flight, b) the width of the moat, c) the rock's velocity at impact.
To solve this problem, we need to break it down into smaller steps. Let's go through it step by step:
Step 1: Understand the given information
- The initial velocity of the rock is 12 m/s at an angle of 25° above the horizon.
- The rock is thrown from atop the castle's outer wall at a height of 9.5 m.
- The rock just clears the surrounding moat.
Step 2: Convert the given information
To make calculations easier, we need to convert the given angle from degrees to radians. This can be done by multiplying the angle by π/180.
Given angle = 25°
Angle in radians = 25° * π/180 ≈ 0.436 radians
Step 3: Determine the rock's time of flight (T)
The time of flight can be calculated using the formula:
T = (2 * initial vertical velocity) / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Initial vertical velocity (Vy) can be found using the initial velocity and the given angle.
Vy = initial velocity * sin(angle)
T = (2 * Vy) / g
Substituting the given values:
T = (2 * 12 m/s * sin(0.436)) / 9.8 m/s²
Step 4: Calculate the width of the moat
To calculate the width of the moat, we need to determine the horizontal distance traveled by the rock during its time of flight. This can be done using the formula:
Horizontal distance (X) = initial horizontal velocity * time
The initial horizontal velocity (Vx) can be found using the initial velocity and the given angle.
Vx = initial velocity * cos(angle)
Substituting the given values:
X = (12 m/s * cos(0.436)) * T
Step 5: Find the rock's velocity at impact
The velocity at impact can be calculated using the horizontal and vertical velocity components at the end of the time of flight. The final velocity is given by:
Vf = √(Vx² + Vy²)
To calculate Vx and Vy, we can use the initial velocity and angle.
Vx = initial velocity * cos(angle)
Vy = initial velocity * sin(angle)
Substituting the given values:
Vf = √((12 m/s * cos(0.436))² + (12 m/s * sin(0.436))²)
Now, let's plug in the values and calculate each part:
Step 3: Determine the rock's time of flight (T)
T = (2 * 12 m/s * sin(0.436)) / 9.8 m/s²
T ≈ 1.89 seconds
Step 4: Calculate the width of the moat
X = (12 m/s * cos(0.436)) * 1.89 s
X ≈ 21.9 meters
Step 5: Find the rock's velocity at impact
Vf = √((12 m/s * cos(0.436))² + (12 m/s * sin(0.436))²)
Vf ≈ 11.8 m/s
Finally:
a) The rock's time of flight is approximately 1.89 seconds.
b) The width of the moat is approximately 21.9 meters.
c) The rock's velocity at impact is approximately 11.8 m/s.
To solve this problem, we can break it down into three parts.
Part a: Finding the time of flight of the rock.
The time of flight refers to the total time it takes for the rock to travel from being thrown to the point it hits the ground. To calculate this, we need to use the horizontal motion of the rock since it is not affected by gravity.
1. First, we need to find the horizontal component of the initial velocity. We can use the given information that the rock is thrown at 12 m/s at an angle of 25° above the horizon.
- The horizontal component of the velocity, Vx, can be calculated using the formula Vx = V * cos(θ), where V is the initial velocity and θ is the angle above the horizon.
- Substituting the values, Vx = 12 m/s * cos(25°).
2. Now, we can use the horizontal component of velocity to calculate the time of flight. The horizontal distance traveled (representing the width of the moat) can be determined using the formula: distance = velocity * time.
- The horizontal distance traveled is equal to the width of the moat.
- Setting the distance equal to the horizontal component of velocity multiplied by the time, we get the equation: distance = Vx * t.
- Substituting the known values, moat_width = (12 m/s * cos(25°)) * t.
3. We can solve for time, t. Rearrange the equation above to isolate t:
- t = moat_width / (12 m/s * cos(25°)).
Part b: Finding the width of the moat.
We already found the time of flight in part a using the known variables. Now, we can substitute the time into the equation used to find the horizontal distance traveled.
1. Use the equation from part a: moat_width = (12 m/s * cos(25°)) * t.
- Substitute the value of time, t, determined earlier.
Part c: Finding the velocity at impact.
To find the velocity at impact, we need to consider both the horizontal and vertical motion of the rock. We can use the equations of kinematics for horizontal and vertical motion.
1. The horizontal velocity remains constant throughout the motion.
- The horizontal component of the velocity, Vx, from earlier is still valid.
2. The vertical velocity changes over time due to the force of gravity.
- The initial vertical velocity, Vy, can be calculated using the formula Vy = V * sin(θ), where V is the initial velocity and θ is the angle above the horizon.
- Substitute the given values to get Vy = 12 m/s * sin(25°).
3. At impact, the rock's vertical displacement is -9.5 m (negative because it is below the starting point).
- Use the formula: displacement = initial_velocity * time + (1/2) * acceleration * time^2.
- The initial vertical velocity, Vy, from step 2 is used in the formula.
- Substitute the known values and rearrange the equation to solve for time.
4. Once we have the time at impact, substitute it into the horizontal velocity formula to get the velocity at impact (the magnitude of the velocity remains the same, but the direction will change when hitting the ground).
By following these steps, you can find the answer to each part of the problem.
To solve this problem, we can break it down into two separate motion problems: the horizontal motion and the vertical motion of the rock.
a) To determine the rock's time of flight, we can focus on the vertical motion. The rock is thrown with an initial velocity of 12 m/s at an angle of 25° above the horizon. The vertical component of the initial velocity can be calculated as follows:
Vertical component of initial velocity = 12 m/s * sin(25°) = 5.14 m/s
Since the rock is thrown vertically upwards and clears the surrounding moat, it will reach its highest point when it has zero velocity. From this highest point, it will then fall downwards. The time of flight is equal to the time it takes for the rock to reach its highest point and then fall back down.
To find the time it takes to reach the highest point, we can use the equation of motion:
Final velocity (v) = Initial velocity (u) + acceleration (a) * time (t)
Since the final velocity at the highest point is zero and the acceleration is due to gravity (g = 9.8 m/s^2), the equation becomes:
0 = 5.14 m/s - 9.8 m/s^2 * t
Solving for t, we get:
t = 0.524 seconds (rounded to 3 decimal places)
Therefore, the rock's time of flight is approximately 0.524 seconds.
b) To determine the width of the moat, we need to calculate the horizontal distance traveled by the rock. This can be done by focusing on the horizontal motion of the rock.
The horizontal component of the initial velocity can be calculated as follows:
Horizontal component of initial velocity = 12 m/s * cos(25°) = 10.87 m/s
The distance traveled by the rock can be calculated using the equation:
Distance (d) = Initial velocity (u) * time (t)
Substituting the known values, we get:
Distance (d) = 10.87 m/s * 0.524 s = 5.69 meters (rounded to 2 decimal places)
Therefore, the width of the moat is approximately 5.69 meters.
c) To determine the rock's velocity at impact, we can combine the horizontal and vertical components of velocity at the moment of impact.
At the highest point, the vertical component of velocity is zero. The horizontal component of velocity remains constant throughout the motion. Therefore, at the moment of impact, the rock's velocity will only have a horizontal component.
The horizontal component of velocity is:
Horizontal component of velocity = 10.87 m/s
Therefore, the rock's velocity at impact is approximately 10.87 m/s, in the horizontal direction.