Gayle runs at a speed of 9.00 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 20.0 m. After she has descended a vertical distance of 4.00 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? (b) What is the sled’s speed just before her brother jumps on? (c) What is the speed of the sled just after the brother jumps on? (d) What is their speed at the bottom of the hill?

Gayle’s mass is 70.0 kg, the sled has a mass of 2.00 kg, and her brother has a mass of 50.0 kg

m=2 kg , m1=70+2=72 kg, m2= 50 kg

v =9 m/s, Δh=4 m, h=20 m
(a)
m1•v =(m1+m)v1
v1= m1/(m1+m)
(b) KE1+ΔPE=KE2
(m1+m)•v1²/2+(m1+m)•g•Δh=(m1+m)•v2²/2
v2=sqrt(v1²+2g•Δh)
(c) (m1+m)•v2= (m1+m2+m) • v3
v3 = (m1+m)•v2/(m1+m2+m)
(d) (m1+m2+m)•v3²/2+ (m1+m2+m)g(h- Δh) = (m1+m2+m)v4²/2
v4=sqrt{ v3²/2+2 g(h- Δh)}.

Can you give me the values you got so I can check mine?

To answer these questions, we can use conservation of mechanical energy and apply the principles of work and energy. Let's break down each question step by step:

(a) What is the sled's speed just after jumping on the sled?

To find the sled's speed just after her brother jumps on, we can use the principle of conservation of mechanical energy. The initial kinetic energy of Gayle and the sled on the hill can be expressed as:

KE_initial = m1 * v_initial^2 / 2

where m1 is the mass of Gayle and the sled (70.0 kg + 2.00 kg) and v_initial is Gayle's initial speed (9.00 m/s).

As they reach a vertical distance of 4.00 m, the potential energy is converted into kinetic energy. The kinetic energy at this point can be calculated as:

KE_m4 = (m1 + m2) * v_m4^2 / 2

where m2 is the mass of Gayle's brother (50.0 kg), and v_m4 is the sled's speed when they descend 4.00 m.

Since there is no energy loss due to friction or other factors mentioned, the total mechanical energy is conserved, and the initial kinetic energy is equal to the kinetic energy at position m4:

m1 * v_initial^2 / 2 = (m1 + m2) * v_m4^2 / 2

Substituting the values, we can solve for v_m4:

(70.0 kg + 2.00 kg) * (9.00 m/s)^2 / 2 = (70.0 kg + 2.00 kg + 50.0 kg) * v_m4^2 / 2

Simplifying the equation:

72.00 * 9.00^2 = 122.00 * v_m4^2

648.00 = 122.00 * v_m4^2

v_m4^2 = 648.00 / 122.00

v_m4^2 = 5.32

Taking the square root of both sides:

v_m4 ≈ √5.32

v_m4 ≈ 2.31 m/s

Therefore, the sled's speed just after Gayle's brother jumps on is approximately 2.31 m/s.

(b) What is the sled's speed just before her brother jumps on?

To find the sled's speed just before Gayle's brother jumps on, we need to consider the change in potential energy and kinetic energy. The potential energy at a vertical distance of 4.00 m is converted into kinetic energy. Since there is no change in mass, the sled's speed just before her brother jumps on is equal to the speed just after jumping on, which is 2.31 m/s.

(c) What is the speed of the sled just after her brother jumps on?

As calculated in question (a), the speed of the sled just after her brother jumps on is approximately 2.31 m/s.

(d) What is their speed at the bottom of the hill?

To find their speed at the bottom of the hill, we can use conservation of mechanical energy again. At the bottom of the hill, the entire potential energy is converted into kinetic energy. The kinetic energy at the bottom can be calculated using:

KE_bottom = (m1 + m2) * v_bottom^2 / 2

Since the sled and Gayle's brother are at rest initially, their initial kinetic energy is zero. This means the initial potential energy is equal to the potential energy at the bottom:

PE_initial = PE_bottom
m1 * g * h_initial = (m1 + m2) * g * h_bottom

where g is the acceleration due to gravity (approximately 9.8 m/s^2), h_initial is the initial vertical drop (20.0 m), and h_bottom is the vertical distance at the bottom of the hill (0 m).

Simplifying the equation:

(70.0 kg + 2.00 kg) * 9.8 m/s^2 * 20.0 m = (70.0 kg + 2.00 kg + 50.0 kg) * 9.8 m/s^2 * 0 m

72.00 * 9.8 * 20.0 = 122.00 * 9.8 * 0

14112 = 0

This equation is not possible because it results in an equality that is not true. In this case, it is not possible to calculate the speed at the bottom of the hill with the given information.

Therefore, we cannot determine their speed at the bottom of the hill with the information provided.