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Convert the given temperatures from celsius to kelvin since we are dealing with gas.
To convert to kelvin, add 273.15 to the temperature in celsius.
T1 = 22 + 273.15 = 295.15 k
T2 = 4 + 273.15 = 277.15 k
V1 = 0.5 L
Let's find the final volume (V2).
To solve for V2 apply Charles Law formula below:
Answer:
The final velocity of the thrower is
and the final velocity of the catcher is
.
Explanation:
Given:
The mass of the thrower,
.
The mass of the catcher,
.
The mass of the ball,
.
Initial velocity of the thrower, 
Final velocity of the ball, 
Initial velocity of the catcher, 
Consider that the final velocity of the thrower is
. From the conservation of momentum,

Consider that the final velocity of the catcher is
. From the conservation of momentum,

Thus, the final velocity of thrower is
and that for the catcher is
.
Answer:
a) K = 2/3 π G m ρ R₁³ / R₂
, b) U = - G m M / r
Explanation:
The law of universal gravitation is
F = G m M / r²
Part A
Let's use Newton's second law
F = m a
The acceleration is centripetal
a = v² / R₂
G m M / R₂² = m v² / R₂
v² = G M / R₂
They give us the density of the planet
ρ = M / V
V = 4/3 π R₁³
M = ρ V
M = ρ 4/3 π R₁³
v² = 4/3 π G ρ R₁³ / R₂
K = ½ m v²
K = ½ m (4/3 π G ρ R₁³ / R₂)
K = 2/3 π G m ρ R₁³ / R₂
Part B
Potential energy and strength are related
F = - dU / dr
∫ dU = - ∫ F. dr
The force was directed towards the center and the vector r outwards therefore there is an angle of 180º between the two cos 180 = -1
U- U₀ = G m M ∫ dr / r²
U - U₀ = G m M (- r⁻¹)
We evaluate for
U - U₀ = -G m M (1 /
- 1 /
)
They indicate that for ri = ∞ U₀ = 0
U = - G m M / r