What type heat transfer is used to fry an egg in a pan and how can you tell

The main type and the dominant one over such a process is the conduction process which is marked with yellow. Conduction heat transfer happens when heat is transferred through solid medium by vibrating its atoms so that the heat energy transfer from one atom to the neighboring atom. This exactly what happens at the pan as it is heated then transfers the heat to the oil then to the egg.
The other way which is has an effect but not significant is the convection heat transfer which happens in liquid medium (ex; air and water) this happens when the hot particles of the liquid that is close to the heat source gets hot and so the volume increases so the density decreases and at last these hot particles rise up and other cold particles take its place and so on.
And this happens, marked in red, at the top layer of the egg in the pan.

Hope this helps.

    m₁v₁ + m₂v₂ = (m₁ + m₂)vmomentum is conserved.the left-hand side of the equation represents the momentum in the system before the collision.the right-hand side represents the momentum in the system after the collisionm₁ and v₁ represents the mass and the velocity of the physics blockm₂ and v₂ represents the mass and velocity of the ball of puttyv represents the final velocity of the combined mass.the question states the two masses stick together. they combined mass has its own velocity with the body mass the sum of the two objects' massestheir combined momentum is (m₁ + m₂)vto find the magitude of (m₁ + m₂)v, we need to add the vectors on the left-hand side of the equation: m₁v₁ + m₂v₂.note the diagram. i have positioned the vectors in accordance with their directions.the magnitude of the resultant can be calculated using the pythagorean theorem since this is a right triangle situation (north and west are perpendicular)|(m₁ + m₂)v| =  √[ (m₁v₁)² + (m₂v₂)² ]|(m₁ + m₂)v| =  √[ (1.5kg·8.0m/s)² + (0.40kg·20m/s)² ]|(m₁ + m₂)v| =  14.4 kg m/sfor two significant figures (assuming that 20m/s is two), this rounds to 14 kg m/s