At what distance will the dog catch up with the cat if the distance between them is 30 meters? - briefly
The distance at which the dog will catch up with the cat can be calculated by considering their respective speeds. If the dog runs twice as fast as the cat, it will close the initial 30-meter gap in 15 seconds.
At what distance will the dog catch up with the cat if the distance between them is 30 meters? - in detail
To determine at what distance the dog will catch up with the cat, given an initial separation of 30 meters, we need to consider the speeds of both animals and the time it takes for the dog to close this gap.
Let's denote:
- ( v_d ) as the speed of the dog,
- ( v_c ) as the speed of the cat,
- ( d ) as the initial distance between them, which is 30 meters.
The relative speed at which the dog is gaining on the cat is given by the difference in their speeds: [ v_{rel} = v_d - v_c ]
Assuming the dogs and cats maintain constant speeds, we can calculate the time ( t ) it takes for the dog to catch up with the cat using the formula: [ t = \frac{d}{v_{rel}} ]
Substituting the given distance ( d = 30 \, \text{meters} ): [ t = \frac{30}{v_d - v_c} ]
During this time, both animals will have covered some distance. The total distance covered by each animal is:
- Distance covered by the dog: ( d_d = v_d \cdot t )
- Distance covered by the cat: ( d_c = v_c \cdot t )
The additional distance the dog travels to catch up with the cat is: [ d_{add} = d_d - d ]
Substituting ( t ) from earlier and simplifying, we get: [ d_{add} = v_d \cdot \frac{30}{v_d - vc} - 30 ] [ d{add} = \frac{30 v_d}{v_d - v_c} - 30 ]
For a concrete example, let's assume the dog runs at ( 10 \, \text{m/s} ) and the cat at ( 5 \, \text{m/s} ): [ v{rel} = 10 - 5 = 5 \, \text{m/s} ] [ t = \frac{30}{5} = 6 \, \text{seconds} ] [ d{add} = 10 \cdot 6 - 30 = 60 - 30 = 30 \, \text{meters} ]
So, in this scenario, the dog will catch up with the cat after an additional 30 meters of running. This calculation can be adjusted for different speeds to determine the exact distance at which the dog will catch up with the cat.