Its all about energy.
While the mass slips down its gravitational energy turns into kinetic energy, till it reaches the end of the slope. This part of the motion is conservative (no friction) which means the total energy is conserved.
Then the mass travels along a path where friction is not neglected anymore which means it loses energy. The energy lost during its process is only the work of the friction force (which is easy to calculate considering the assumptions made).
Before hitting the wall, the mass has to compress the spring entirely, which means it has to transform some of its kinetic energy into elastic energy. So the energy initially put in the system (due to the gravitational potential) minus the energy lost by friction has to be greater than the energy needed to press the spring, in order for the mass to hit the wall.
After that, the mass will bounce and a similar reasoning will give you the answer for b. But be careful, as it is not obvious that the mass will have enough energy left to reach the slope.
do you need help to write the expressions of these different energies ?
While the mass slips down its gravitational energy turns into kinetic energy, till it reaches the end of the slope. This part of the motion is conservative (no friction) which means the total energy is conserved.
Then the mass travels along a path where friction is not neglected anymore which means it loses energy. The energy lost during its process is only the work of the friction force (which is easy to calculate considering the assumptions made).
Before hitting the wall, the mass has to compress the spring entirely, which means it has to transform some of its kinetic energy into elastic energy. So the energy initially put in the system (due to the gravitational potential) minus the energy lost by friction has to be greater than the energy needed to press the spring, in order for the mass to hit the wall.
After that, the mass will bounce and a similar reasoning will give you the answer for b. But be careful, as it is not obvious that the mass will have enough energy left to reach the slope.
do you need help to write the expressions of these different energies ?