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What are the speeds of two objects if, w...

What are the speeds of two objects if, when they move uniformly towards each other, they get 4 m closer in each second and when they move uniformly in the same direction with the original speeds, they get 4 m closer each 10 s?

A

`2.8 m//s` and `1.2 m//s`

B

`5.2 m//s` and `4.6 m//s`

C

`3.2 m//s` and `2.1 m//s`

D

`2.2 m//s` and `1.8 m//s`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two scenarios given for the two objects moving towards each other and moving in the same direction. ### Step 1: Define the Variables Let: - \( v_1 \) = speed of the first object (in m/s) - \( v_2 \) = speed of the second object (in m/s) ### Step 2: Analyze the First Scenario In the first scenario, the two objects are moving towards each other. The relative speed when they move towards each other is given by the sum of their speeds: \[ v_1 + v_2 = 4 \, \text{m/s} \] This is because they get 4 meters closer to each other every second. ### Step 3: Write the First Equation From the first scenario, we can write our first equation: \[ v_1 + v_2 = 4 \quad \text{(Equation 1)} \] ### Step 4: Analyze the Second Scenario In the second scenario, the two objects are moving in the same direction. The relative speed when they move in the same direction is given by the difference of their speeds: \[ v_1 - v_2 = \frac{4 \, \text{m}}{10 \, \text{s}} = 0.4 \, \text{m/s} \] This is because they get 4 meters closer to each other every 10 seconds. ### Step 5: Write the Second Equation From the second scenario, we can write our second equation: \[ v_1 - v_2 = 0.4 \quad \text{(Equation 2)} \] ### Step 6: Solve the Equations Now we have a system of two equations: 1. \( v_1 + v_2 = 4 \) 2. \( v_1 - v_2 = 0.4 \) We can add these two equations to eliminate \( v_2 \): \[ (v_1 + v_2) + (v_1 - v_2) = 4 + 0.4 \] This simplifies to: \[ 2v_1 = 4.4 \] Now, divide both sides by 2: \[ v_1 = 2.2 \, \text{m/s} \] ### Step 7: Find \( v_2 \) Now we can substitute \( v_1 \) back into Equation 1 to find \( v_2 \): \[ 2.2 + v_2 = 4 \] Subtract \( 2.2 \) from both sides: \[ v_2 = 4 - 2.2 = 1.8 \, \text{m/s} \] ### Final Answer The speeds of the two objects are: - \( v_1 = 2.2 \, \text{m/s} \) - \( v_2 = 1.8 \, \text{m/s} \)
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