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Sonja [21]
3 years ago
15

A projectile is shot directly away from Earth's surface. Neglect the rotation of the Earth. What multiple of Earth's radius RE g

ives the radial distance (from the Earth's center) the projectile reaches if (a) its initial speed is 0.241 of the escape speed from Earth and (b) its initial kinetic energy is 0.241 of the kinetic energy required to escape Earth. (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?
Physics
2 answers:
Elden [556K]3 years ago
8 0

Answer:

a. 1.06R b. 1.32R c. GMm/R

Explanation:

a. Considering the conservation of mechanical energy,

K₁ + U₁ = K₂ + U₂     (1)where K₁ and K₂ = initial and final kinetic energies of projectile, and U₁ and U₂ = initial and final potential energies of projectile

K₁ = 1/2mv²,   K₂ = 0,  U₁ = -GMm/R   where R = radius of earth , U₂ = '-GMm/r where r = radius at maximum height.

So, inputting the variables into (1), we have

1/2mv² - GMm/R = 0 - GMm/r

1/2mv² = -GMm/r + GMm/R  (2)

Now v = 0.241v₁    where v₁ =√(2GM/R) escape velocity

Substituting v into (2) above, we have

1/2m[0.241√(2GM/R)]² = -GMm/r + GMm/R

0.058GMm/R = -GMm/r + GMm/R

GMm/r = GMm/R -'0.058GMm/R

GMm/r = 0.942GMm/R

r = R/0.942

r = 1.06R

b. When K₁ = 0.241K wh6ere K = escape kinetic energy = GMm/R. So

K₁ + U₁ = K₂ + U₂

0.241GMm/R -'GMm/R = 0 -' GMm/r

-'0.759GMm/R = '-GMm/r

r = R/0.759 = 1.32R

c. If it is to escape earth, its initial velocity must equal the escape velocity.

So its least initial mechanical energy is its escape kinetic energy

1/2m[√(2GM/R)]² = GMm/R

7nadin3 [17]3 years ago
3 0

Answer:

(a) r = 1.062·R_E = \frac{531}{500} R_E

(b) r = \frac{33}{25} R_E

(c) Zero

Explanation:

Here we have escape velocity v_e given by

v_e =\sqrt{\frac{2GM}{R_E} } and the maximum height given by

\frac{1}{2} v^2-\frac{GM}{R_E} = -\frac{GM}{r}

Therefore, when the initial speed is 0.241v_e we have

v = 0.241\times \sqrt{\frac{2GM}{R_E} } so that;

v² = 0.058081\times {\frac{2GM}{R_E} }

v² = {\frac{0.116162\times GM}{R_E} }

\frac{1}{2} v^2-\frac{GM}{R_E} = -\frac{GM}{r} is then

\frac{1}{2} {\frac{0.116162\times GM}{R_E} }-\frac{GM}{R_E} = -\frac{GM}{r}

Which gives

-\frac{0.941919}{R_E} = -\frac{1}{r} or

r = 1.062·R_E

(b) Here we have

K_i = 0.241\times \frac{1}{2} \times m \times v_e^2 = 0.241\times \frac{1}{2} \times m  \times \frac{2GM}{R_E} = \frac{0.241mGM}{R_E}

Therefore we put  \frac{0.241GM}{R_E} in the maximum height equation to get

\frac{0.241}{R_E} -\frac{1}{R_E} =-\frac{1}{r}

From which we get

r = 1.32·R_E

(c) The we have the least initial mechanical energy, ME given by

ME = KE - PE

Where the KE = PE required to leave the earth we have

ME = KE - KE = 0

The least initial mechanical energy to leave the earth is zero.

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3 years ago
A real object is 10.0 cm to the left of a thin, diverging lens having a focal length of magnitude 16.0 cm. What is the location
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Answer:

A)6.15 cm to the left of the lens

Explanation:

We can solve the problem by using the lens equation:

\frac{1}{q}=\frac{1}{f}-\frac{1}{p}

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q is the distance of the image from the lens

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p is the distance of the object from the lens

In this problem, we have

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If the ball is 0.60 mm from her shoulder, what is the tangential acceleration of the ball? This is the key quantity here--it's a
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This question is incomplete, the complete question is;

In a softball windmill pitch, the pitcher rotates her arm through just over half a circle, bringing the ball from a point above her shoulder and slightly forward to a release point below her shoulder and slightly forward. (Figure 1) shows smoothed data for the angular velocity of the upper arm of a college softball pitcher doing a windmill pitch; at time t = 0 her arm is vertical and already in motion. For the first 0.15 s there is a steady increase in speed, leading to a final push with a greater acceleration during the final 0.05 s before the release. In each part of the problem, determine the corresponding quantity during the first 0.15 s of the pitch.

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a) What is the angular acceleration?

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Given the data in the question;

from the graph below;

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Angular acceleration ∝ = 12 / 0.15

Angular acceleration ∝ = 80 rad/s²

Therefore, the angular acceleration is 80 rad/s²

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the tangential acceleration of the ball will be;

a = ∝ × s

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a = 80 × 0.6

a = 48 m/s²

a = ( 48 / 9.8 )g

a = 4.9 g

Therefore, the tangential acceleration of the ball is;

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Answer:

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The velocity of the wave in the string is given by :

v=f\lambda\\\\v=234\times 0.31\\\\v=72.54\ m/s

So, the velocity of the waves in the string is 72.54 m/s.

3 0
3 years ago
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