To tackle this problem, we need to analyze the motion of the stones thrown by Dan and Ned from the top of the tower. Since they are throwing the stones horizontally, we can break down the motion into horizontal and vertical components. Let’s go through each part step by step.
Understanding the Motion
Both stones are thrown horizontally, which means their initial vertical velocity is zero. The only force acting on them vertically is gravity, which will cause them to fall downwards. The height of the tower is 19.6 meters, and we will use this information to determine how long it takes for the stones to hit the ground.
Part A: Time Taken for Velocity Vectors to Become Perpendicular
To find the time when the velocity vectors of the two stones become perpendicular, we first need to understand their horizontal and vertical velocities. The horizontal velocities are constant since there is no horizontal acceleration (ignoring air resistance). Dan's stone has a horizontal velocity of 1 m/s, and Ned's stone has a horizontal velocity of 4 m/s.
The vertical velocity of both stones increases due to gravity. The vertical velocity at any time \( t \) can be calculated using the formula:
- Vertical Velocity \( v_y = g \cdot t \)
where \( g = 10 \, \text{m/s}^2 \). Therefore, at time \( t \), the vertical velocities will be:
- Dan's stone: \( v_{y1} = 10t \)
- Ned's stone: \( v_{y2} = 10t \)
For the velocity vectors to be perpendicular, the dot product of their velocity vectors must equal zero. The velocity vectors can be represented as:
- Dan's velocity vector: \( \vec{v_1} = (1, 10t) \)
- Ned's velocity vector: \( \vec{v_2} = (4, 10t) \)
The dot product is given by:
- Dot Product \( \vec{v_1} \cdot \vec{v_2} = 1 \cdot 4 + (10t) \cdot (10t) = 4 + 100t^2 \)
Setting the dot product to zero for perpendicularity:
This equation has no real solution, indicating that the horizontal components of the velocities will never allow the vectors to be perpendicular. Therefore, the time taken for the velocity vectors to become perpendicular is:
- **Time = Infinity** (they never become perpendicular)
Part B: Horizontal Distance Between the Two Stones When Their Velocity Vectors Are Perpendicular
Since we established that the velocity vectors never become perpendicular, we cannot calculate a horizontal distance at that moment. However, we can calculate the horizontal distance each stone travels until they hit the ground.
First, we find the time taken for each stone to fall 19.6 meters. Using the formula for vertical motion:
- \( h = \frac{1}{2} g t^2 \)
Substituting the values:
- \( 19.6 = \frac{1}{2} \cdot 10 \cdot t^2 \)
- \( 19.6 = 5t^2 \)
- \( t^2 = \frac{19.6}{5} = 3.92 \)
- \( t = \sqrt{3.92} \approx 1.98 \, \text{s} \)
Now, we can find the horizontal distances:
- Dan's horizontal distance: \( d_1 = 1 \cdot 1.98 \approx 1.98 \, \text{m} \)
- Ned's horizontal distance: \( d_2 = 4 \cdot 1.98 \approx 7.92 \, \text{m} \)
The horizontal distance between the two stones when they hit the ground is:
- **Distance = 7.92 m - 1.98 m = 5.94 m**
Part C: Time Taken for Displacement Vectors to Become Perpendicular
For the displacement vectors to be perpendicular, we can analyze their positions at any time \( t \). The displacement vectors can be represented as:
- Dan's displacement: \( \vec{d_1} = (1t, -\frac{1}{2}gt^2) = (1t, -5t^2) \)
- Ned's displacement: \( \vec{d_2} = (4t, -\frac{1}{2}gt^2) = (4t, -5t^2) \)
For the displacement vectors to be perpendicular, the dot product must also equal zero:
- \( \vec{d_1} \cdot \vec{d_2} = (1t)(4t) + (-5t^2)(-5t^2) = 4t^2 + 25t^4 = 0 \)
Since \( t^2 \) cannot be negative, this equation also has no real solution, indicating that the displacement vectors will never be perpendicular. Thus, the time taken for the displacement vectors to become perpendicular is:
- **Time = Infinity** (they never become perpendicular)
Summary of Results
To summarize:
- A. Time for velocity vectors to become perpendicular: **Infinity**
- B. Horizontal distance when they hit the ground: **5.94 m**
- C. Time for displacement vectors to become perpendicular: **Infinity**
This analysis shows how the horizontal and vertical motions interact and how they can lead to different outcomes in terms of perpendicularity. If you have any further questions or need clarification on any part, feel free to ask!
