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The total energy TE = mgh + 1/2 mU^2; where h = 20 m, g = 9.81 m/sec^2, and U = 10 mps. When the ball reaches max height H, all that TE will be potential energy PE = mgH = TE.
So there you are. TE = mgh + 1/2 mU^2 = mgH = TE from the conservation of energy. Solve for H.
1) H = (gh + 1/2 U^2)/g = h + U^2/2g = ? meters where everything on the RHS is given. You can do the math.
2) As the ball drops from H to h, it picks up KE as the potential energy mgH is converted when the potential energy is diminished to mgh, where h < H. So PE - pe = ke = mg(H - h) = 1/2 mv^2 so solve for v = sqrt(2g(H - h)) and, again, everything is given. You can do the math.
3) Same deal as 2) except now its V = sqrt(2gH) because all the PE = mgH = 1/2 mV^2 = KE when it is about to hit the ground. You can do the math.
Both matter and energy can eneter and exit an <span>Open System only. A.</span>
Answer:
v = 9.04 m / s
Explanation:
For this exercise we can use the relation that the work of the non-conservative force (friction) is equal to the variation of the mechanical energy of the system.
W = Em_f - Em₀ (1)
Starting point. Lower slope
Em₀ = K = ½ m v²
highest point. Where is the skier at a height h
Em_f = U = m g h
The work of rubbing
W = -fr x
the negative sign is because the friction force opposes the movement.
Let's set a reference system where the x axis is parallel to the slope and the y axis is perpendicular
let's use trigonometry to break down the weight
cos θ = W_y / W
sin θ = Wₓ / W
W_y = W cos θ
Wₓ = W sin θ
Y axis
N - Wₓ = 0
N = mg sin θ
X axis
fr = m a
the friction force has the expression
fr = μ N
fr = μ mg sin θ
we look for the job
W = - μ mg sin θ x
where x is the distance along the slope
we substitute in 1
-μ mg sin θ x = mg h - ½ m v²
let's use trigonometry to find the distance x
tan 30 = h / x
x = h / tan 30
we substitute
- 
= m gh - ½ m v²
we use
tan 30 = sin30 / cos30
v² = 2g h + 2 μ g h cos 30
v = 
let's calculate
v = 
v = 9.04 m / s
Answer: A.
Explanation: Roughly 180 - 200 million years ago, just before the first dinosaurs evolved. Mammals themselves evolved from a group or reptiles which exhibited mammal-like traits. One of them was specialized teeth. Reptiles tend to have teeth all the same shape. The mammal-like reptiles evolved tiny teeth in front of the jaw and two pairs of over sized fangs along the the sides. Like modern mammals, the head was large in proportion to the rest of the body. The jaws were also evolving another mammal trait, the ability to move sideways. Despite the lack of specialized teeth, acute hearing and the ability to chew, the dinosaurs evolved an adaptation which made them far more successful than mammals--modified leg bones. These limbs could be articulated directly under their bodies. This enabled the legs to support more weight, since the limbs were now under the body instead of at the sides. Then dinosaurs did something which secured their dominance for the next 120 million years - they began to stand on two legs. Although the back was still parallel to the ground, running on two legs greatly increased the dinosaur's speed. Mammals could simply not compete with swift, giant predators and were forced to remain small, and most became nocturnal to evade dinosaurs which were probably active during the day. Despite that they managed to survive which allowed the further evolution of mammals into us, humans.
