Gayle runs at a speed of 8.00 m/s and dives on a sled, initially at rest on the top of a frictionless, snow-covered hill, that has a vertical drop of 14.0 m. After she has descended a vertical distance of 5.00 m, her brother, who is initially at rest, hops on her back, and they continue down the hill together. (a) What is the sled’s speed just after jumping on the sled?


Gayle’s mass is 47.0 kg, the sled has a mass of 6.00 kg, and her brother has a mass of 35.0 kg.

7.09 m/s

To find the sled's speed just after Gayle's brother jumps on the sled, we can use the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.

Before Gayle's brother jumps on the sled, Gayle has a mass of 47.0 kg and a velocity of 8.00 m/s, while the sled has a mass of 6.00 kg and is initially at rest. Gayle's brother, who is initially at rest, has a mass of 35.0 kg.

The total momentum before Gayle's brother jumps on the sled can be calculated as:
Total momentum before = (Gayle's mass × Gayle's velocity) + (sled's mass × sled's velocity) + (brother's mass × brother's velocity)
Total momentum before = (47.0 kg × 8.00 m/s) + (6.00 kg × 0 m/s) + (35.0 kg × 0 m/s)
Total momentum before = 376.0 kg⋅m/s

After Gayle's brother jumps on the sled, they continue down the hill together. The total mass of the system (Gayle, the sled, and her brother) is now 47.0 kg + 6.00 kg + 35.0 kg = 88.0 kg.

Let's assume the sled's speed just after Gayle's brother jumps on is v_sled.

Using the conservation of momentum principle, we can equate the total momentum before and after the event:
Total momentum before = Total momentum after
376.0 kg⋅m/s = (total mass × sled's speed after)

Rearranging the equation, we can solve for the sled's speed after jumping on:
sled's speed after = (total momentum before) / (total mass)
sled's speed after = 376.0 kg⋅m/s / 88.0 kg
sled's speed after = 4.27 m/s

Therefore, the sled's speed just after Gayle's brother jumps on is approximately 4.27 m/s.