To find the mean velocity of the point over the entire distance, we need to break down the problem into manageable parts. Let's denote the total distance traveled as \( D \). According to the problem, the point travels half of this distance, \( \frac{D}{2} \), with a velocity \( v_0 \). For the remaining half of the distance, the point covers it in two segments: the first half of the time at velocity \( v_1 \) and the second half at velocity \( v_2 \).
Step-by-Step Breakdown
1. Calculate the time taken for the first half of the distance
The time taken to travel the first half of the distance, \( \frac{D}{2} \), with velocity \( v_0 \) can be calculated using the formula:
Time for first half (t0): \( t_0 = \frac{\frac{D}{2}}{v_0} = \frac{D}{2v_0} \)
2. Analyze the second half of the distance
For the second half of the distance, we need to consider how the time is divided between the two velocities. Let's denote the total time taken to cover the second half of the distance as \( T \). According to the problem, this time is split equally between \( v_1 \) and \( v_2 \), meaning:
- Time at \( v_1 \): \( \frac{T}{2} \)
- Time at \( v_2 \): \( \frac{T}{2} \)
3. Calculate the distance covered in the second half
The distance covered at each velocity can be expressed as follows:
- Distance at \( v_1 \): \( d_1 = v_1 \cdot \frac{T}{2} \)
- Distance at \( v_2 \): \( d_2 = v_2 \cdot \frac{T}{2} \)
Since the total distance for the second half is \( \frac{D}{2} \), we can set up the equation:
Equation: \( d_1 + d_2 = \frac{D}{2} \)
Substituting the expressions for \( d_1 \) and \( d_2 \) gives:
\( v_1 \cdot \frac{T}{2} + v_2 \cdot \frac{T}{2} = \frac{D}{2} \)
Factoring out \( \frac{T}{2} \) leads to:
\( \frac{T}{2} (v_1 + v_2) = \frac{D}{2} \)
From this, we can solve for \( T \):
Time for second half (T): \( T = \frac{D}{v_1 + v_2} \)
4. Total time of motion
The total time \( T_{total} \) for the entire journey is the sum of the time for the first half and the second half:
Total time: \( T_{total} = t_0 + T = \frac{D}{2v_0} + \frac{D}{v_1 + v_2} \)
5. Mean velocity calculation
The mean velocity \( V_{mean} \) is defined as the total distance divided by the total time:
Mean velocity: \( V_{mean} = \frac{D}{T_{total}} \)
Substituting \( T_{total} \) into this equation gives:
\( V_{mean} = \frac{D}{\frac{D}{2v_0} + \frac{D}{v_1 + v_2}} \)
We can simplify this expression by factoring out \( D \):
\( V_{mean} = \frac{1}{\frac{1}{2v_0} + \frac{1}{v_1 + v_2}} \)
To combine the fractions, we find a common denominator:
\( V_{mean} = \frac{1}{\frac{(v_1 + v_2) + 2v_0}{2v_0(v_1 + v_2)}} \)
Thus, the mean velocity can be expressed as:
Final formula: \( V_{mean} = \frac{2v_0(v_1 + v_2)}{(v_1 + v_2) + 2v_0} \)
This formula gives you the mean velocity of the point averaged over the entire time of motion, taking into account the different velocities and the time spent at each. Understanding this concept is crucial in physics, especially in kinematics, where velocity and time play significant roles in motion analysis.
