a cube slides down a frictionless incline as shown and inelastically strikes a cube at the battom of the incline that's 1/2 (one-half) it's mass. The coeff. of restitution between the two masses is 0.7. If incline is 30 cm high and table is 90cm off floor, at what Horizontal positions do each of the cubes land with respect to the edge of table?

To solve this problem, let's break it down step-by-step:

Step 1: Calculate the velocity of the first cube at the bottom of the incline.
Since the incline is frictionless, the potential energy at the top of the incline is converted entirely into kinetic energy at the bottom. The conservation of energy equation is as follows:
m1 * g * h = 0.5 * m1 * v^2, where m1 is the mass of the first cube, g is the acceleration due to gravity, h is the height of the incline, and v is the velocity of the first cube at the bottom.
Plugging in the values, we can solve for v:
m1 * 9.8 * 0.3 = 0.5 * m1 * v^2
2.94 = 0.5 * v^2
v^2 = 2.94 / 0.5
v^2 = 5.88
v = sqrt(5.88)
v ≈ 2.42 m/s

Step 2: Calculate the velocity of the second cube at the bottom of the incline.
Since the two cubes collide and stick together inelastically, we can use the law of conservation of momentum to calculate the velocity after the collision.
m1 * v = (m1 + m2) * vf, where vf is the final velocity of the combined cubes.
Plugging in the values, we can solve for vf:
m1 * 2.42 = (m1 + 0.5m1) * vf
2.42 = 1.5 * vf
vf = 2.42 / 1.5
vf ≈ 1.61 m/s

Step 3: Calculate the horizontal distance each cube lands from the edge of the table.
The time it takes for both cubes to fall from the incline to the floor is the same since they start at the same height. We can use the equations of motion to calculate the horizontal distances.
For the first cube:
d1 = v * t, where d1 is the horizontal distance traveled by the first cube, v is the velocity of the first cube at the bottom, and t is the time it takes to reach the bottom.
The vertical distance traveled by the first cube is the height of the incline:
h = 0.5 * g * t^2
0.3 = 0.5 * 9.8 * t^2
0.3 = 4.9 * t^2
t^2 = 0.3 / 4.9
t^2 ≈ 0.061
t ≈ sqrt(0.061)
t ≈ 0.247 s

Plugging in the values, we can solve for d1:
d1 = 2.42 * 0.247
d1 ≈ 0.598 m

For the second cube:
d2 = vf * t, where d2 is the horizontal distance traveled by the second cube, vf is the velocity of the combined cubes, and t is the time it takes to reach the bottom.
Plugging in the values, we can solve for d2:
d2 = 1.61 * 0.247
d2 ≈ 0.397 m

Step 4: Calculate the total horizontal distance.
The total horizontal distance is the sum of the distances traveled by each cube.
Total distance = d1 + d2
Total distance ≈ 0.598 + 0.397
Total distance ≈ 0.995 m

Therefore, the first cube lands approximately 0.598 m from the edge of the table, and the second cube lands approximately 0.397 m from the edge of the table.

To determine the horizontal positions where each of the cubes land with respect to the edge of the table, we need to analyze the physical principles involved in the scenario.

Let's break down the problem step by step:

1. Calculate the vertical velocities of each cube just before impact:
- Given that the coefficient of restitution (COR) between the two masses is 0.7, the vertical velocity of the first cube just before impact will be equal to 0.7 times the velocity it had just before falling.
- The second cube has no initial vertical velocity because it is at rest.

2. Calculate the time it takes for the first cube to reach the bottom of the incline:
- Using kinematic equations, we can find the time it takes for the first cube to fall from a height of 30 cm.
- The formula to calculate the time of fall is t = √(2h/g), where h is the height and g is the acceleration due to gravity (approximately 9.8 m/s²).

3. Calculate the horizontal distance covered by the first cube during its fall:
- The horizontal distance covered is given by the formula d = v * t, where v is the horizontal velocity of the first cube, and t is the time calculated in step 2.
- The horizontal velocity of the first cube remains constant since there is no horizontal force acting on it after it leaves the incline.

4. Calculate the horizontal distance covered by the second cube during the same time period:
- Since the second cube is at rest until it is hit by the first cube, it remains at a constant height of 90 cm above the floor.
- Using the formula d = v0 * t + 0.5 * a * t², where v0 is the initial vertical velocity (0 m/s), t is the time calculated in step 2, and a is the acceleration due to gravity (approximately -9.8 m/s² because it acts in the opposite direction), we can find the displacement or distance covered by the second cube.

5. Add the distance covered by the second cube to its initial distance from the edge of the table to get the final position.

By following these steps and plugging in the given values, you can calculate the horizontal positions where each of the cubes land with respect to the edge of the table.