Question

A 3.0-kg block (A) is attached to a 1.0-kg block (B) by a massless spring that is compressed and locked in place, as shown in the figure. The blocks slide without friction along the x-direction at an initial constant speed of =2.0 m/s. At time 0=0 s, the positions of blocks A and B are xA,i=1.0 m and xB,i=1.4 m, respectively, at which point a mechanism releases the spring, and the blocks begin to oscillate as they slide.

If 2.0 s later block B is located at xB,f=7.1 m, what will be the position xA,f of block A?
Determine the final speed of the first stone 1,f.

Determine the final direction of the second stone as an angle relative to the first stone's initial direction of motion. Let counterclockwise from the first stone's initial direction define the direction of positive angles

Answer 1

The final position of block A is 6.7 m. The final speed of the first block is -2.85 m/s. The final direction of the second block is 45° relative to the first block's initial direction of motion.

Step-by-step explanation:

To determine the position xA,f of block A, we need to find the displacement of block B first. Since block B was initially located at xB,i=1.4 m and 2.0 s later it is at xB,f=7.1 m, the displacement of block B is given by:

ΔxB = xB,f - xB,i = 7.1 m - 1.4 m = 5.7 m

Since the blocks are attached by a massless spring, the displacement of block A will be the negative of the displacement of block B:

ΔxA = -ΔxB = -5.7 m

Therefore, the final position of block A, xA,f, can be found by subtracting the displacement from the initial position:

xA,f = xA,i - ΔxA = 1.0 m - (-5.7 m) = 6.7 m

The final speed of the first block, 1,f, can be found by dividing the displacement by the time:

1,f = ΔxA / Δt = -5.7 m / 2.0 s = -2.85 m/s

Since the initial speed is positive in the positive x-direction, the final speed will be negative. Therefore, the magnitude of the final speed of the first block is 2.85 m/s in the opposite direction to its initial motion. The final direction of the second block relative to the first block's initial direction of motion can be determined by taking the inverse tangent of the displacement ratio:

θ = arctan(ΔxB / ΔxA) = arctan(-5.7 m / -5.7 m) = arctan(1) = 45°