a cyclist approaches a 5m hill with a speed of 11m/s. how much work is required to get up the hill?

What speed do you wish to go over the top at?

Clearly you cannot allow the speed to fall to zero or the speed would not be the only thing falling.

However from the speed I KNOW that you will crest the hill without even pedalling with energy left over.

From the numbers

Ek = 1/2 m v^2

energy needed = mgh

1/2 m * 11 ^ 2 - m * 9.8 * 5 = 1/2 * m * v2^2

60.5 - 49 = 1/2 v2^2

11.5 = 1/2 v2^2

v2 = sqrt( 2 * 11.5) = 4.8 m/s or about 17 km/hr

WITHOUT adding any work at all.

(1/2)mv^2 = m*60.5 m^2/s^2;

mgh = m*49 m^2/s^2.

The extra work needed to get the cyclist and bike to the top of the hill will be:

their combined mass times (11.5 m^2/s^2).