A rope is stretched between two boats at rest. A sailor in the first boat
pulls the rope with a constant force of 100 N. First boat with the sailor
has a mass of 250 kg whereas the mass of second boat is double of
this mass. If the initial distance between the boats was 30 m, the time
taken for two boats to meet each other is (neglect water resistance
between boats and water)
To find the time taken for the two boats to meet, we can use the concept of relative motion.
First, we need to determine the acceleration of the two boats. To do this, we'll use Newton's second law of motion:
F = ma
Where:
F = force applied by the sailor (100 N)
m = mass of the boat (250 kg for the first boat and 2 * 250 kg = 500 kg for the second boat)
a = acceleration of the boat
For the first boat:
100 N = 250 kg * a1
a1 = 100 N / 250 kg
a1 = 0.4 m/s^2
For the second boat:
100 N = 500 kg * a2
a2 = 100 N / 500 kg
a2 = 0.2 m/s^2
Now, we can calculate the time taken for the two boats to meet. The equation to use here is the equation of motion:
s = ut + 0.5at^2
Where:
s = initial distance between the boats (30 m)
u = initial velocity of the second boat (0 m/s)
a = relative acceleration of the two boats
t = time taken for the two boats to meet
Rearranging the equation, we get:
t = sqrt(2s / a)
For the relative acceleration, we'll take the difference between the two boat's accelerations since they are moving in opposite directions:
a = a1 - a2
a = 0.4 m/s^2 - 0.2 m/s^2
a = 0.2 m/s^2
Substituting the values into the equation, we have:
t = sqrt(2 * 30 m / 0.2 m/s^2)
t = sqrt(60 m / 0.2 m/s^2)
t = sqrt(300 s^2)
t ≈ 17.32 s
Therefore, the time taken for the two boats to meet each other is approximately 17.32 seconds.