Respuesta :
To solve this problem, we'll use principles of mechanics, specifically the concepts of work, energy, and motion of the center of mass.
1. **Calculate the force with which the plane was pushed:**
The work done by the constant force on the inclined plane is equal to the change in kinetic energy of the plane.
\[Work = \Delta KE = \frac{1}{2} Mv_{\text{final}}^2 - \frac{1}{2} Mv_{\text{initial}}^2\]
Given that the plane was pushed for 5 seconds and then left free to slide, the initial velocity of the plane, \(v_{\text{initial}}\), is zero. We're also given the mass of the plane, \(M\), as 1 kg.
\[Work = \frac{1}{2} \times 1 \times v_{\text{final}}^2 - \frac{1}{2} \times 1 \times 0^2 = \frac{1}{2} \times 1 \times v_{\text{final}}^2\]
The work done on the plane is also equal to the force applied multiplied by the distance it moves during the 5 seconds.
\[Work = Force \times \text{distance}\]
The distance moved by the plane during the 5 seconds depends on its acceleration, which we can calculate using the equations of motion:
\[s = ut + \frac{1}{2}at^2\]
Since \(u = 0\) (initial velocity), the distance moved simplifies to:
\[s = \frac{1}{2}at^2\]
Given \(t = 5\) seconds and \(s\) is the distance moved by the plane, we can solve for \(a\), the acceleration of the plane during the pushing phase.
2. **Calculate the speed with which the ball leaves the inclined plane:**
We'll use the principle of conservation of energy to find the speed of the ball at the top of the inclined plane. The potential energy gained by the ball at the top is equal to the kinetic energy it had at the bottom, neglecting any energy losses due to friction.
3. **Calculate the speed with which the plane continues to move after the ball has left it:**
We can use the principle of conservation of momentum to determine the speed of the plane after the ball leaves it. Initially, the center of mass of the system (plane + ball) is at rest. After the ball leaves, the center of mass continues to move with the same velocity, but now only the plane contributes to it.
4. **Determine the motion of the center of mass of the system:**
As mentioned above, initially, the center of mass is at rest. After the ball leaves, the center of mass continues to move with a velocity determined by the conservation of momentum.
Let me know if you need further explanation or help with calculations for any specific part!
1. **Calculate the force with which the plane was pushed:**
The work done by the constant force on the inclined plane is equal to the change in kinetic energy of the plane.
\[Work = \Delta KE = \frac{1}{2} Mv_{\text{final}}^2 - \frac{1}{2} Mv_{\text{initial}}^2\]
Given that the plane was pushed for 5 seconds and then left free to slide, the initial velocity of the plane, \(v_{\text{initial}}\), is zero. We're also given the mass of the plane, \(M\), as 1 kg.
\[Work = \frac{1}{2} \times 1 \times v_{\text{final}}^2 - \frac{1}{2} \times 1 \times 0^2 = \frac{1}{2} \times 1 \times v_{\text{final}}^2\]
The work done on the plane is also equal to the force applied multiplied by the distance it moves during the 5 seconds.
\[Work = Force \times \text{distance}\]
The distance moved by the plane during the 5 seconds depends on its acceleration, which we can calculate using the equations of motion:
\[s = ut + \frac{1}{2}at^2\]
Since \(u = 0\) (initial velocity), the distance moved simplifies to:
\[s = \frac{1}{2}at^2\]
Given \(t = 5\) seconds and \(s\) is the distance moved by the plane, we can solve for \(a\), the acceleration of the plane during the pushing phase.
2. **Calculate the speed with which the ball leaves the inclined plane:**
We'll use the principle of conservation of energy to find the speed of the ball at the top of the inclined plane. The potential energy gained by the ball at the top is equal to the kinetic energy it had at the bottom, neglecting any energy losses due to friction.
3. **Calculate the speed with which the plane continues to move after the ball has left it:**
We can use the principle of conservation of momentum to determine the speed of the plane after the ball leaves it. Initially, the center of mass of the system (plane + ball) is at rest. After the ball leaves, the center of mass continues to move with the same velocity, but now only the plane contributes to it.
4. **Determine the motion of the center of mass of the system:**
As mentioned above, initially, the center of mass is at rest. After the ball leaves, the center of mass continues to move with a velocity determined by the conservation of momentum.
Let me know if you need further explanation or help with calculations for any specific part!
