Two cars leave the same point at the same time, each travelling at the same constant speed of 89.0 km/h and each having the same mass. However, the first car drives directly east, while the second car heads directly south. At what speed (the magnitude of the velocity) is the centre of mass of these two cars travelling? Give your answer as a positive value in km/h (without a sign).
To find the speed of the center of mass of the two cars, we need to calculate the resultant velocity vector.
Since the first car is driving directly east and the second car is driving directly south, their velocities are perpendicular to each other. We can represent the velocity of the first car as v1 = 89.0 km/h to the right (positive x-direction) and the velocity of the second car as v2 = 89.0 km/h downward (negative y-direction).
The center of mass velocity is given by the vector sum of v1 and v2. To find the magnitude of this resultant velocity vector, we can use the Pythagorean theorem:
resultant velocity = √(v1^2 + v2^2)
Substituting the given values:
resultant velocity = √((89.0 km/h)^2 + (89.0 km/h)^2)
Calculating:
resultant velocity = √((7921 km^2/h^2 + 7921 km^2/h^2)
resultant velocity = √(15842 km^2/h^2)
resultant velocity ≈ 125.9 km/h
Therefore, the speed of the center of mass of the two cars is approximately 125.9 km/h.