Question: You have an apple tree that produces high-quality apples in large numbers. The tree is also resistant to a number of h
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Answer:
He should run at least at 1.5 m/s
The diver will enter the water at an angle of 87° below the horizontal.
Explanation:
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The position and velocity of the diver are given by the following vectors:
r = (x0 + v0x · t, y0 + v0y · t + 1/2 · g · t²)
v = (v0x, v0y + g · t)
Where:
r = position vector at time t
x0 = initial horizontal position
v0x = initial horizontal velocity
t = time
y0 = initial vertical position
v0y = initial vertical velocity
g = acceleration due to gravity (-9.8 m/s² considering the upward direction as positive)
v = velocity vector at time t
Please, see the attached figure for a description of the problem. Notice that the origin of the frame of reference is located at the jumping point so that x0 and y0 = 0.
We know that, to clear the rocks, the position vector r final (see figure) should be:
r final = ( > 5.0 m, -50 m)
So let´s find first at which time the y-component of the vector r final is - 50 m:
y = y0 + v0y · t + 1/2 · g · t²
-50 m = 2.1 m/s · t - 1/2 · 9.8 m/s² · t²
0 = -4.9 m/s² · t² + 2.1 m/s · t + 50 m
Solving the quadratic equation
t = 3.4 s
Now, we can calculate the initial horizontal velocity using the equation of the x-component of the position vector knowing that at t =3.4 the horizontal component should be greater than 5.0 m:
x = x0 + v0x · t (x0 = 0)
5.0 m < v0x · 3.4 s
v0x > 5.0 m / 3.4 s
v0x > 1.5 m/s
The initial horizontal velocity should be greater than 1.5 m/s
To find the angle at which the diver enters the water, we have to find the magnitude of the final velocity (vector vf in the figure). We already know the magnitude of the x-component of the vector vf, since the horizontal velocity is constant. So:
vfx > 1.5 m/s
Now, let´s calculate vfy:
vfy = v0y + g · t
vfy = 2.1 m/s - 9.8 m/s² · 3.4 s
vfy = -31 m/s
Let´s calculate the minimum magnitude that the final velocity will have if the diver safely clears the rocks. Let´s consider the smallest value allowed for vfx: 1.5 m/s. Then:
Then the final velocity of the diver will be greater or equal than 31 m/s.
To find the angle, we have to use trigonometry. Notice in the figure that the vectors vf, vfx and vy form a right triangle in which vf is the hypotenuse, vfx is the adjacent side and vfy is the opposite side to the angle. Then:
cos θ = adjacent / hypotenuse = vfx / vf = 1.5 m/s / 31 m/s
θ = 87°
The diver will enter the water at an angle of 87° below the horizontal.
Answer:
19.68 × 10⁻³ m
Explanation:
Given;
Original Length, L₁ = 41.0 m
Temperature Change, ΔT = 40.0°C
Thermal Linear expansion of steel is given to be, ∝ = 12 × 10⁻⁶ /°C
Generally, Linear expansivity is expressed as;
∝ = ΔL / L₁ΔT
Where
∝ is the Linear expansivity
ΔL is the change in length, L₂ - L₁
L₂ is the final length
L₁ is the original length
ΔT is the change in temperature θ₂ - θ₁ (Final Temperature - Initial Temperature)
From equation of linear expansivity
ΔL = ∝L₁ΔT
ΔL = 12 × 10⁻⁶ /°C × 41.0 m × 40.0 °C
ΔL = 19.68 × 10⁻³ m
ΔL = 19.68 mm
Answer:
v = 28.98 ft / s
Explanation:
For this problem we must solve it in parts, let's start by looking for the speed of the two cars after the collision
In the exercise they indicate the weight of each car
Wₐ = 1500 lb
W_b = 1125 lb
Car B's velocity from v_b = 42.0 mph westward, car A travels east
let's find the mass of the vehicles
W = mg
m = W / g
mₐ = Wₐ / g
m_b = W_b / g
mₐ = 1500/32 = 46.875 slug
m_b = 125/32 = 35,156 slug
Let's reduce to the english system
v_b = 42.0 mph (5280 foot / 1 mile) (1h / 3600s) = 61.6 ft / s
We define a system formed by the two vehicles, so that the forces during the crash have been internal and the moment is preserved
we assume the direction to the east (right) positive
initial instant. Before the crash
p₀ = mₐ v₀ₐ - m_b v_{ob}
final instant. Right after the crash
p_f = (mₐ + m_b) v
the moment is preserved
p₀ = p_f
mₐ v₀ₐ - m_b v_{ob} = (mₐ + m_b) v
v =
we substitute the values
v =
v = 0.559 v₀ₐ - 26.40 (1)
Now as the two vehicles united we can use the relationship between work and kinetic energy
the total mass is
M = mₐ + m_b
M = 46,875 + 35,156 = 82,031 slug
starting point. Jsto after the crash
K₀ = ½ M v²
final point. When they stop
K_f = 0
The work is
W = - fr x
the negative sign is because the friction forces are always opposite to the displacement
Let's write Newton's second law
Axis y
N-W = 0
N = W
the friction force has the expression
fr = μ N
we substitute
-μ W x = Kf - Ko
-μ W x = 0 - ½ (W / g) v²
v² = 2 μ g x
v = Ra (2 0.750 32 17.5
v = 28.98 ft / s