Hematoma or Contusion: This is the most likely diagnosis given the history of trauma. A direct blow to the back of the knee can cause bleeding into the soft tissues, leading to a hematoma or contusion, which presents as pain, swelling, and bruising.

Baker's Cyst Rupture: Trauma can cause a pre-existing Baker's cyst to rupture, leading to pain and swelling in the back of the knee.
Muscle Strain: The muscles at the back of the knee, such as the hamstring or popliteus, can be strained due to the trauma, causing pain and limited mobility.
Ligament Sprain: The ligaments around the knee, including those at the back, can be sprained due to trauma, leading to instability and pain.

Popliteal Artery Injury: Although less common, a severe trauma can cause injury to the popliteal artery, which runs at the back of the knee. This is a medical emergency that requires prompt intervention to prevent limb loss.
Deep Vein Thrombosis (DVT): Trauma can increase the risk of DVT, especially if there is significant swelling or immobilization. DVT can be life-threatening if it leads to a pulmonary embolism.
Fracture: Certain fractures, such as a tibial plateau fracture or a fracture of the fibula, can present with pain at the back of the knee after trauma.

Ganglion Cyst: Although rare, a ganglion cyst can form at the back of the knee and cause pain, especially if it becomes large enough to compress nearby structures.
Tumors: Tumors, either benign or malignant, can cause pain at the back of the knee, but this would be extremely rare as a direct result of trauma.
Infection: In rare cases, trauma can introduce bacteria into the tissues, leading to an infection such as cellulitis or abscess formation at the back of the knee.