When two bodies move uniformly towards each other, the distance decrease by 6 m/s. If both the bodies moves (as above) in the same direction with the same speed, the distance between them increases by 4 m/s. Then the speed of the two bodies are :
5 m/s, 1 m/s
Calculating Speeds Using Relative Velocity
This problem involves the concept of relative velocity, which is how the velocity of one body appears to an observer on another body.
Let the speeds of the two bodies be \(v_1\) and \(v_2\). We are given information about how the distance between them changes under two different scenarios.
Scenario 1: Bodies Moving Towards Each Other
When two bodies move towards each other, the rate at which the distance between them decreases is the sum of their individual speeds. This is their relative speed.
Given that the distance decreases by 6 m/s, we can write the first equation:
\[v_1 + v_2 = 6 \quad (1)\]
Scenario 2: Bodies Moving in the Same Direction
When two bodies move in the same direction, the rate at which the distance between them changes is the absolute difference between their individual speeds. This is their relative speed.
Given that the distance increases by 4 m/s, this implies the faster body is moving away from the slower body. Let's assume \(v_1 > v_2\). The relative speed is then \(v_1 - v_2\).
We can write the second equation:
\[v_1 - v_2 = 4 \quad (2)\]
Solving the System of Equations
Now we have a system of two linear equations with two variables \(v_1\) and \(v_2\):
\[v_1 + v_2 = 6\]
\[v_1 - v_2 = 4\]
We can solve this system by adding the two equations together:
\[(v_1 + v_2) + (v_1 - v_2) = 6 + 4\]
\[v_1 + v_2 + v_1 - v_2 = 10\]
\[2v_1 = 10\]
\[v_1 = \frac{10}{2}\]
\[v_1 = 5 \text{ m/s}\]
Now substitute the value of \(v_1\) into either equation. Using equation (1):
\[5 + v_2 = 6\]
\[v_2 = 6 - 5\]
\[v_2 = 1 \text{ m/s}\]
So, the speeds of the two bodies are 5 m/s and 1 m/s.
Verification
- If speeds are 5 m/s and 1 m/s and they move towards each other, relative speed = 5 + 1 = 6 m/s. Distance decreases by 6 m/s. Correct.
- If speeds are 5 m/s and 1 m/s and they move in the same direction (faster one ahead), relative speed = 5 - 1 = 4 m/s. Distance increases by 4 m/s. Correct.
The speeds of the two bodies are 5 m/s and 1 m/s.
Important Questions from Relative Velocity
Rain is falling vertically on the ground at speed 5√3 m/s. If a man walks towards the East with speed 5 m/s, he will feel the rain falling at what angle to the vertical ?
When a stone thrown directly upwards reaches the top, it:
A. Velocity and acceleration are zero.
B. The velocity is zero and the acceleration is about 10 m/s 2.
C. Velocity is about 10 m/s and acceleration is zero.
D. The velocity is about 10 m/s and the acceleration remains the same.
On a rainy day, rain drops fall vertically with speed \(v\). An observer is moving towards east with speed \(v\), while another observer moves towards north with the same speed \(v\). What is the angle between apparent rain directions observed by them?
