Two cars leave the same point at the same time, each travelling at the same constant speed and each having the same mass. The first car drives directly east, while the second car heads directly north. If the centre of mass of the two-car system is travelling at 51 km/h (i.e. the magnitude of the velocity), how fast is each car travelling in km/h?
The CM moves diagonally NE at a speed that is the individual car speed multiplied by sqrt2. The individual car speed is therefore 51*0.707 = 36 km/h
53
To solve this problem, we can use vector addition. Let's define the velocities of the two cars as vectors:
v1 = velocity of the first car (going east)
v2 = velocity of the second car (going north)
Now, we know that the magnitude of the velocity of the center of mass is 51 km/h. The center of mass is determined by the average of the positions of the two cars at any given time, so we can write:
|v_cm| = √(v1^2 + v2^2)
To find the speed of each car (v1 and v2), we need to work with the components of their velocities. Since the first car is going east and the second car is going north, we have:
v1 = v1x i (east direction)
v2 = v2y j (north direction)
Now, we can use the Pythagorean theorem for vectors to find the magnitudes of the velocity components:
v1^2 = v1x^2 + v1y^2
v2^2 = v2x^2 + v2y^2
Note that the mass of the cars is not relevant in this problem, as we are only concerned with their velocities.
Since the cars have the same mass and are leaving from the same point at the same time, we can assume that the time they have been traveling is equal. Let's say they have been traveling for t hours.
Now, we have the following information:
|v_cm| = 51 km/h
v1 = v1x i
v2 = v2y j
t = travel time of both cars (same for both cars)
Since we know that the magnitude of the total velocity of the center of mass is equal to 51 km/h, we can equate the two equations:
51^2 = (v1x + v2x)^2 + (v1y + v2y)^2
Now, notice that the components of the velocities are in a right-angled triangle, and the magnitudes satisfy the Pythagorean theorem. From this, we can deduce that the angles these velocities make with the x-axis and y-axis are complementary angles.
Using this, we can use trigonometry to express the components of the velocities:
v1x = cos(θ) * v1
v1y = sin(θ) * v1
v2x = cos(θ) * v2
v2y = sin(θ) * v2
Substituting these expressions back into the equation:
51^2 = (cos(θ) * v1 + cos(θ) * v2)^2 + (sin(θ) * v1 + sin(θ) * v2)^2
Expanding, we get:
51^2 = (v1^2 + v2^2 + 2 * v1 * v2 * cos(θ)) + (v1^2 + v2^2 + 2 * v1 * v2 * sin(θ))
Combining like terms:
51^2 = 2 * (v1^2 + v2^2) + 2 * v1 * v2 * (cos(θ) + sin(θ))
Solving for v1 and v2, we have:
v1^2 + v2^2 + v1 * v2 * (cos(θ) + sin(θ)) = 51^2 / 2
Now, this equation doesn't give us enough information to solve for v1 and v2 precisely because we have two unknowns. We need additional information such as the angle θ or the value of either v1 or v2 to find specific values.
Therefore, without more information, we cannot determine the exact speed of each car (v1 and v2).