Gayle runs at a speed of 4.00 m/s and dives on a sled, initially
atrest on the top of a frictionless, snow-covered hill. aftershe
has descended a vertical distance of 5.00m, her brother, who
isinitially at rest, hops on her back, and they continue down
thehill together.
gayle's mass is 50.0kg, the sled's mass is 5.00kg and her
brotherhas a mass of 30.0kg.

what is their speed at the bottom of the hill if the total
verticaldrop is 15.0m?

16.58 m/s

Explanation:

Let g = 10 m/s2

The total (kinetic and potential) energy of Gayle at top of the hill (15 m) is

where = 50 + 5 = 55 kg is the mass of Gayle and the sledge, = 4m/s is the speed of Gayle on top, = 15 m is the vertical height of Gayle on top.

Similarly, the potential energy of her brother at 15 - 5 = 10m high is:

where = 30kg is the mass of the brother, = 10 m is the vertical height of the brother when he jumps on.

The total energy is:

According to energy conservation, all of this is converted to kinetic energy at the bottom:

where M = 50 + 30 + 5 = 85 kg is the total system mass. V is the system mass at the bottom

0.11 m/s and 0.28 m/s should be correct

the total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)