The **mass **of the objects in **kilograms **could be obtained as; 100 Kg and 16.3 Kg.

The term **weight **has to do with the effect of the** gravitational force** on an object hat is found on the surface of the earth. Note that the weight is a force an d the direction of the weight must always be towards the center of the earth.

Now we are asked to convert 981N and 160N to **kilograms**. Recall that the weight is the product of the mass of the object in kilograms to the acceleration due to gravity.

Mass = weight/acceleration due to gravity

1) 981N/9.8 m/s^2

= 100 Kg

2) 160N/9.8 m/s^2

= 16.3 Kg

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What is the distance from the moon to the sun?

**Answer:**

About 150 million kilometers

**Explanation:**

have good day♡

Around 1500 kilometers or 2 suns away could also say 932.057 miles away

A horse pulls a cart with force F. As a result of this force the cart accelerates with constant acceleration. The

magnitude of the force that the cart exerts on the horse

A) is zero newtons. B) greater than the magnitude of F. C) equal to the magnitude of F. D).

less than the magnitude of F.

**Answer:**

c. equal equal to the magnitude of f

Exercise write the peice wise

function {h(x) x

r(x) a<=x

K(x) x>=b} in terms

Of one function

A **piecewise**-defined **function **in mathematics is one that is composed of several **smaller functions**, each of which has a **certain interval **of the **domain **it applies to. Instead of being a property of the function itself, piecewise definition is a means to express the **function**.

In terms of **r(x),**

**r(x) <= x when x >= b**

A function that is defined by many **sub**-**functions**, each of which applies to a **specific interval **of the main **function's domain**, is referred to as a **piecewise-defined function** in mathematics (also known as a piecewise function or a **hybrid function**) (a sub-domain).

When an input value passes specific "**boundaries**," a rule or relationship changes. In these cases, we employ piecewise functions to explain the situation.

For instance, it happens frequently in business that the price per piece of a particular item gets decreased after the quantity bought exceeds a certain threshold.

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Mr. Red Herring was found shot dead in his backyard. He was about a foot away from his back porch, lying next to his personal handgun. A bloody footprint was also found on the porch. Analysis of a bullet found nearby suggests that, based on the striation marks, the bullet that killed Herring came from a gun discarded in a nearby trash can. He apparently had a date that night with Mrs. Scarlet. Based on various evidence, the forensics team also created a digital rendering of the crime.

An example of **demonstrative evidence** is the **digital rendering** of the crime scene that shows where the gun was fired from. That is **option B.**

A **demonstrative evidence** is the type of evidence that is being represented using visuals to help enhance the facts of a claim made against an opponent in the law court.

The components of **demonstrative evidence** include the following:

Based on the various evidence presented concerning the sudden death of Mr. Red Herring who was short dead in his backyard, the **demonstrative evidence** is when the **forensics** team also created a digital rendering of the crime showing where the gun was shot.

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**Options of question**:

In this story, which is an example of demonstrative evidence?

The fingerprint that showed that Mrs. Scarlet handled the gun

The digital rendering of the crime scene that shows where the gun was fired from

The report about bullet striations that prove the shot came from the discarded weapon

The DNA evidence gathered from the blood splatters

A mass on the end of a spring undergoes simple harmonic motion. At the instant when the mass is at its equilibrium position, what is its instantaneous acceleration?.

A mass on the end of a spring undergoes simple harmonic motion. At the instant when the mass is at its equilibrium position, its **instantaneous acceleration is zero**

When the restoring **force** is proportionate to the displacement but acting in the opposite direction, an oscillating mass moves in a manner known as **harmonic motion**. The sine wave has a constant frequency and amplitude and can be used to describe **harmonic motion** because it is periodic. The motion of a weight on a spring is an illustration of this.

Simple Harmonic Motion, also known as **SHM**, is a motion in which the restoring** force** is inversely correlated with the body's deviation from its mean position. A particle moving in simple harmonic motion experiences acceleration given by the formula a(t) = − 2 x (t). The particle's angular velocity is given in this equation as.

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1. on the planet arrakis a male ornithoid is flying toward his mate at 25.0 m/s while singing at a frequency of 1200 hz. if the stationary female hears a tone of 1240 hz, what is the speed of sound in the atmosphere of arrakis?

If the stationary female hears a tone of 1240 **hertz**, then the speed of the sound in the atmosphere of arrakis will be 775 meter per second.

The term "Ducler shift" refers to the apparent change in **frequency **caused by the source moving closer to the observer or by the relative motion between the source and the object. The formula: true **frequency **over 1 minus v s over v gives the observed frequency. Let's call this equation 1, shall we? The real frequency in our case is stated as 1200 **hertz**, but the perceived frequency is reported as 1240 hertz. It is stated that the source's velocity, vs, is 25.0 meters per second. Rearranging equation 1 yields the result that the real **frequency **over the absorbed is equal to 1 minus b s over v.

As a result, we can be represented as v s over 1 minus f, actual over f absorbed now 1 substituting the values we get 25.0 meter per and over 1 minus 1200 hertz over 1240 **hertz**. Frequency fur, the rearranging we get 1 minus f real over f absorbed, is equal to v s over v. The sound travels at a speed of 775 meters per second after further calculation.

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Light travels at a speed of about 3.0 108 m/s.(a) How many miles does a pulse of light travel in a time interval of 0.1 s, which is about the blink of an eye?Δx = mi(b) Compare this distance to the diameter of Earth. (Use 6.38 106 m for the radius of the Earth.)ΔxDE =

Given:

Speed of light = 3 x 10⁸ m/s

Let's solve for the following:

• (a). How many miles does a pulse of light travel in a time interval of 0.1 s, which is about the blink of an eye?

Apply the formula:

[tex]\Delta x=v*t[/tex]Where:

v is the speed of light

t is the time.

Thus, we have:

[tex]\begin{gathered} \Delta x=3.0\times10^8*0.1 \\ \\ \Delta x=3.0\operatorname{\times}10^7\text{ m} \end{gathered}[/tex]Now let's convert the answer from meters to miles.

Where:

1 mile = 1609.34 meters

[tex]\begin{gathered} 3.0\times10^7=\frac{3.0\times10^7}{1609.34} \\ \\ =18641.14\text{ mi} \end{gathered}[/tex]**Δx = 18641.14 mi**

• (b). Compare this distance to the diameter of Earth.

Apply the formula:

[tex]\frac{\Delta x}{D_E}=\frac{\Delta x}{2*r}[/tex]Where:

r = 6.38 x 10⁶ m.

Thus, we have:

[tex]\frac{\Delta x}{D_E}=\frac{3.0\times10^7}{2*6.38\times10^6}=2.35[/tex]**ANSWER:**

• (a). 18641.14 mi

,• (b). 2.35

a conductor consists of an infinite number of adjacent wires, each infinitely long and carrying a current i (whose direction is out-ofthe-page), thus forming a conducting plane.

If there are n wires per unit length, **magnitude** of B measure at A or C is **B = Ωo*n*I/2**

A key concept in science is the concept of **magnitude** in physics. The general quantity or distance is referred to as magnitude. When it comes to the elements of movement, we can tie magnitude to an object's size and motion speed.

A particular object's magnitude is determined by its size or quantity. For instance, when it comes to** speed**, if a car is moving at a quicker rate than a nearby motorcycle, the magnitude of the automobile's speed is greater than the **speed** of the motorcycle. Although they contain directions as well as magnitude, vector quantities. Vector quantities include things like force, acceleration, **speed**, velocity, and many others. A vector's absolute value is referred to as its magnitude.

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two cars are diriving down the road. car a has a mass of 1,100 kg and is moving at 20 m/s. car b has a mass of 1000 kg and is moving at 30 m/s. which car has more kinetic energy and why

**Answer:**

Hope the pictures will help you

describe how two objects can have the same speed but different velocities

Two objects can have the same **speed** but different **velocities**. The detailed description is given below.

We can take two escalators to explain the concept. The two escalators are moving up at a speed of 10 m/s. In this case, the two escalators have the same **speed** and they are moving in the same **direction**. So, they have the same velocity also.

Now one escalator is moving up at a speed of 10 m/s and the other escalator moving down at a speed of 10 m/s. In this case, both escalators move at 10 m/s, so their speed is the same. But they are not moving in the same direction. Hence their **velocity** is not the same. One has 10 m/s downwards and the other one has 10 m/s upwards velocity.

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Since gravity is a conservative force, the work done against gravity can be replaced by gravitational potential energy. When an object is thrown vertically upward, its gravitational potential energy true or false?.

A ball's **gravitational potential energy** will rise as it is launched vertically upward.

An object's** gravitational potential energy** is the power it has as a result of being in a gravitational field. The gravitational potential energy is most frequently used for an object close to the Earth's surface, when the gravitational **acceleration **is believed to remain constant at around 9.8 m/s2. Since any point can be chosen as the gravitational potential energy zero (just like any coordinate system zero), the potential energy at a height h above that point is equal to the work that would be needed to lift the item to that height with no net change in kinetic energy. Its weight must be lifted with the same amount of force, so the gravitational potential energy must also be equal. The gravitational potential energy is most frequently used for an object close to the Earth's surface, when the gravitational **acceleration **is believed to remain constant at around 9.8 m/s2. Multiple factors affect the gravitational potential energy of an object: its mass, the gravitational **acceleration **it experiences from the earth, and its distance from the ground.

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A trumpet plays middle C (262 Hz). How fast would it have to be moving to raise the pitch to C sharp (277 Hz)? Use 343 m/s as the speed of sound.

The** trumpet **would have to be moving at a** speed **of approximately 345.8 m/s to raise the pitch to **C sharp.**

**What is speed?**

In everyday **language **and **kinematics**, an object's **speed **(typically abbreviated as "v") is defined as the size of the change in **position **that occurs over **time **or the amount of change that occurs **per period **of time; it therefore a **vector quantity**. The instantaneous **speed **is the upper limit of the average speed as the **time interval's **duration gets closer to **zero**; it is calculated by dividing the distance travelled by the time **interval's **duration. **Speed **and **velocity **are distinct concepts.

**Speed **is measured as **distance **divided by time.

To raise the pitch of a note by one semitone, the **speed **of sound must be increased by approximately 6%. In this case, the** trumpet **would have to be moving at approximately 364 m/s to raise the pitch to** C sharp.**

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Five ramps lead from the ground to the second floor of a workshop, as sketched below. All five ramps have the same height; ramps B, C, D and E have the same length; ramp A is longer than the other four. You need to push a heavy cart up to the second floor and you may choose any one of the five ramps. Assuming no frictional forces on the cart, which ramp would require you to do the least work?

1. Same work for ramps B, C, D or E; more work for ramp A.

2. Same work for the straight ramps A and B; less work for ramps C, D, and E.

3. Ramp D.

4. Ramp E.

5. Ramp B.

6. Ramp A.

7. Ramp C.

8. Same work for all five ramps.

9. Unable to determine without knowing the exact profiles of ramps C, D or E.

By assuming no fri**ctional forces** on the cart, Same work for all five** ramps** would require you to do the least work.

The **force** produced by two surfaces coming into contact and sliding against one another is referred to as **frictional force**. The following variables impact the frictional force, Surface roughness and the amount of force pressing them together have the biggest an impact on these forces.

According to the **work energy theorem**, the net change in a body's energy is equal to the work that is done on it. K.E. or P.E. Proof: Consider a mass'm' object travelling at a 'u' starting **velocity**. Allow a body to vary its velocity to v in response to a constant **force**, F.

Let h be the height of the **ramps**. Since there is no** friction**, we can use the

work-energy theorem.

Wtot = Wperson + Wgravity = ∆K

Hence for all the ramps,

Wperson = −Wgravity + ∆K = mgh + ∆K

In particular, if ∆K = 0 (the cart starts from rest and ends at rest),

Wperson = mgh, for all the ramps.

So, the Same work for all** five ramps**.

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A wire of a certain length (α = 0.0065 1/°C) has a resistance of 15 Ω at 20°C.

Calculate the temperature at which the resistance will be 22.8 Ω

The **temperature** at which the resistance will be 22.8 Ω is 100 °C

The following data were obtained from the question:

Coefficient of epansion (α) = 0.0065 °C¯¹ Original resistance (R₁) = 15 Ω Original temperature (T₁) = 20 °C New resistance (R₂) = 22.8 ΩThe new temperature can be obtained as illustrated below:

α = R₂ – R₁ / R₁(T₂ – T₁)

0.0065 = 22.8 – 15 / 15(T₂ – 20)

0.0065 = 7.8 / 15(T₂ – 20)

Cross multiply

0.0065 × 15 (T₂ – 20) = 7.8

0.0975 (T₂ – 20) = 7.8

Divide both side by 0.0975

T₂ – 20 = 7.8 / 0.0975

T₂ – 20 = 80

Collect like terms

T₂ = 80 + 20

**T₂ = 100 °C**

Thus, the **temperature** is 100 °C

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A dart shot at a monkey is traveling 10 m/s horizontally, while simultaneously traveling 7 m/s downward. At what angle below the horizontal is the dart traveling? Answer in degrees

The **angle below **the horizontal the dart shot at 10 m/s horizontally and 7 m/s downward is 55°

In a right angled triangle,

**tan θ = Opposite side / Adjacent side**

Here the resultant velocity, horizontal and vertical components makes a right angled triangle.

Opposite side = Horizontal component = 10 m / s

Adjacent side = Vertical component = 7 m / s

tan θ = 10 / 7

tan θ = 1.43

θ = [tex]tan^{-1}[/tex] ( 1.43 )

θ = 55°

The formula used to solve the problem is a **trigonometric ratio**. Trigonometric ratios are applicable only on a right angled triangle. Some of the basic trigonometric ratios are:

Therefore, the angle below the horizontal, the dart is traveling is **55°**

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Two cars are moving along a stright line in the same direction with a velocity of 25 km/h and 30 km/h respectively. find the velocity of car a relative to car b

**Answer:**

**Explanation:**

V = Vb - Va = 30 - 25 = 5 km/h

3. Hakeem was participating in a psychology experiment. For the first ten minutes Hakeem would hear a click then a puff of air would blow in Hakeem's eye. His eyes would blink and tear up. For the second ten minutes Hakeem would hear a click but the puff of air would only blow in his eye every third time. Hakeem knows this but his eyes would blink and tear up after every click.

UCS: UCR:

CS: CR:

NS:

In the **psychology experiment** in which Hakeem participates, UCS is click, UCR is blink and tear up, CS is blowing a puff of air, CR is thought of the puff of air and NS is the click before puff of air was introduced.

In psychology the **abbreviations **of the given terms are as follows:

Here the **Unconditioned Response** is the blinking and tearing up because he does that even though there is no puff of air. Since clicking provokes the Unconditioned Response, clicking is the Unconditioned Stimulus. The clicking is paired with puff of air for multiple times, that is why Unconditioned Response is stimulated. So the puff of air is the Conditioned Stimulus.

Therefore,

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a ball is thrown in the air at 5m/s. If the ball was thrown at a 4 degrees angle. How long will it take to return to its original height?

**Answer:**

0.5seconds

**Explanation:**

where

initial velocity u=5m/s

gravity g=9.8m/s²

final velocity v=0m/s (since velocity at maximum height is 0)

time t=?

using

v=u-gt

0=5-9.8(t)

0=5-9.8t

9.8t=5

t=5/9.8

t=0.5102040816

t=0.5sec

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Jack and Jill carry a bag of bricks to the top of the hill to mend the well. If the bricks weigh 50 N and the hill is

30 m, how much work do they do between them?

**Answer:**

1500 J

**Explanation:**

Work = force * distance = 50 N * 30 M = 1500 J

for the following trajectory, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. r(t), for 0

The Fundamental Theorem of Calculus is used to evaluate the definite integral. A important tool for simulating the dynamics of moving objects and other physical phenomena is the **parametric **form.

For a parameter t, the dynamics or trajectory of a moving object is given in **parametric **form. We determine the speed of the object over the range of its parameter using derivatives and a definite integral. The Fundamental Theorem of Calculus is used to evaluate the definite integral. A important tool for simulating the dynamics of moving objects and other physical phenomena is the **parametric **form.

The velocity function or vector's absolute value or norm can be used to calculate an object's speed.

We obtain the **velocity **v(t) as the first derivative of the position trajectory given as follows since the velocity is the rate of change of the distance traveled:

v ( t ) = r ′ ( t ) = ⟨ ( 2 t 3 ) ′ , ( − t 3 ) ′ , ( 3 t 3 ) ′ ⟩ = ⟨ 6 t 2 , − 3 t 2 , 9 t 2 ⟩

The speed is given as either its norm or absolute value in the **velocity **function above as follows:

Speed(t)=∥v(t)∥=∥⟨6t2,−3t2,9t2⟩∥

=√(6t2)2+(−3t2)2(9t2)2=√36t4+9t4+81t4=√126t2(2)

The speed at t = 7 will be (2) from (2).

Speed(7)=√126×72=49√126.

Using (2) and the Calculus Fundamental Theorem, the trajectory's length L can be determined as follows:

L = ∫ 7 0 ∥ r ′ ( t )∥ d t = ∫ 7 0 ∥ v ( t ) ∥ d t = ∫ 7 0 √ 126 t 2 d t = √ 126 [ t 3 3 ] 7 0 = √ 126 [ 7 3 3 − 0 3 3 ] = 7 3 √ 126 3. \s.

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A force of -4.4 x 103 N exists between a positive charge of 8.0 x 10-4 C and a negative charge of -3.0 x 10-4 C. What is the distance that separates the charges?

The **distance **that separates the charges is** 0.7 m.**

The **distance **between the charges is determined by applying Coulomb's law of electrostatic force as shown below.

F = kq₁q₂/r²

where;

r is the distance between the chargesk is coulomb's constantF is the force between the chargesr² = kq₁q₂/F

r² = (9 x 10⁹ x 8 x 10⁻⁴ x 3 x 10⁻⁴) / (4.4 x 10³)

r² = 0.49

r = √0.49

r = 0.7 m

Thus, the **distance **that separates the charges is** 0.7 m.**

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inner parts of the flattening cloud begin to fall freely inward, raining down on growing object at the center. gravitational potential energy of collapsing gas cloud is converted into heat and radiative energy.

These 2 statements are 2 steps of **solar planet** formation 1st statement is 3rd step 2nd statement is 4th step.

In the cosmos, there are a lot of planetary systems with **solar planets** around a host star, similar to our own. Because we refer to things that are connected to our star as "solar," our planetary system is sometimes known as "the solar system," after the Latin word for the** Sun,** "solis." The **Sun**, our star, and everything gravitationally connected to it, including the moons, planets, and dwarf planets like Pluto, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, make up our solar system. Thousands of planetary systems orbiting other stars in the Milky Way have been found, in addition to our solar system. The Milky Way galaxy's outer spiral arm is where our solar system is situated. In our solar system, there is only the **Sun**.

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a change in state involves a change in the______of the particles

**Answer: Energy**

**Explanation: Any change in the states of matter is the result of added or lost energy. For example, when Water is frozen, it loses all of its particle energy, which is why when it's ice, all the particles stick together, but if you boiled the water to make vapor, they fly around like crazy because they have a lot of energy.**

according to a simplified model of a mammalian heart, at each pulse approximately 20 g of blood is accelerated from 0.25 m/s to 0.35 m/s during a period of 0.10 s. what is the magnitude of the force exerted by the heart muscle?

The **magnitude** of the **force **exerted by the heart muscle is 0.02N

since

F = m dv/dt

= 0.02(0.35-0.25)/0.10

F =0.02N.

A** force** is an influence in physics that can change the motion of an object. A force can cause a mass object to change its velocity, or accelerate. Intuitively, force can be described as a push or a pull. A force is a vector quantity because it has both **magnitude** and direction.

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as the pressure of an enclosed gas decreases to half its original value, what happens to the volume of the gas if temperature is held constant?(1 point)

**Answer:**

See below

**Explanation:**

P1V1/T1 = P2V2/T2 IF T1 = T2

Then

P1V1 = P2V2 P2 = 1/2 P1

P1V1 = 1/2 P1 V2 DIVIDE BOTH SIDES BY P1

V1 = 1/2 V2 MULTIPLY BOTH SIDES BY 2

2V1 = V2 OR V2 = 2 V1 THE NEW VOLUME IS TWICE ORIGINAL

A car travels south at 30 m/s for 5 minutes. How many seconds does it travel

for?

A. 350 s

B. 250 s

C. 200 s

D. 300 s

**Answer:**

The Answer is D. 300 s because in 5 minutes there are 300 seconds

An object moves 60.0 m on a bearing (angle from North) of 60.0°. If the object then moves 30.0 m North, how far is it from the start point? You may use a scale diagram or trigonometry to answer this question.

**Answer: x(t2)−x(t1) over the time interval [t1,t2]**

**Explanation: hope this helps **

two conductors made of the same material are connected across the same potential difference. conductor a has eight times the diameter and eight times the length of conductor b. what is the ratio of the power delivered to a to the power delivered to b?

The **power **given to A is 7 times more powerful than the power delivered to B. So, the ratio of the power delivered to a to the power delivered to b is 7:1.

A wire's cross sectional area is computed as follows:

A= πd²/4

A wire's resistance is calculated as;

R= pL/A

R= 4pL/πd²

opposition in **wire **A;

R= 4pALA/πd²A

opposition in wire B;

P = V²/R

wired **power **delivery;

P= V²A/RA

energy provided through cable A;

P= V²b/Rb

energy transferred through **wire **B;

Replace R's value in the power delivered through wire A;

PA= V²A/RA = V²Aπd²/4pALA

PA/PB = d²A/LA x LB/V²B/d²B

The diameter and length of wire A are both seven times greater than those of **wire **B;

PA : PB = 7 : 1

Consequently, the **power **given to A divided by the power transferred to B equals 7 : 1

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a parallel-plate capacitor is formed from two 4.0 cm x 4.0 cm electrodes spaced 2.0 mm apart. the electric field strength inside the capacitor is 1.0 x 106 n/c. what is the charge (in nc) on each electrode?

Equation Q/A = E **Electric Field **within a** parallel plate capacitor**

∴ **Q = E*Aε**

E = 10^6 N/C

A = 4*4 = 16cm^2 = 16 * 10^-4 m^2

**ε = 8.852 * 10^-12 F/m**

**Q = +/- 10^6 * 16 * 10^-4 * 8.852 * 10^-12**

= +/- 141.6 * 10^-10 C

**= +/- 14.2 nC**

You can think of an** electric field **as a physical field that surrounds all **charged particles **and pulls on them all.

Before the dielectric breaks down, a** parallel plate capacitor** can only **hold a certain amount of energy.** It can be stated as follows: The parallel plate capacitor is a device that uses two parallel plates connected across a battery to create an electric field between them.

A **nonmetallic component **of a circuit is made in touch with using an **electrode**, an electrical conductor. Depending on the type of battery, electrodes are a crucial component that can be made of a range of materials.

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For questions 7 through 9, Albert and friends are stranded in a clearing on top of a hill at an

elevation of 350m above sea level. You are on a rescue plane flying in supplies to tide them over

until help arrives. Your plane is flying at a constant speed of 260 km/h from west to east at an

elevation of 810m.

PLEASE HELP WOTH THESE QUESTIONS!!!!!!

**Answer:**

Assume that the air resistance on the supplies is negligible, and that [tex]g = 9.8 \; {\rm m\cdot s^{-2}}[/tex].

The plane need to drop the supplies when it is horizontally approximately [tex]700\; {\rm m}[/tex] away from the hill.

The supplies will hit the tree.

**Explanation:**

Let [tex]u_{y}[/tex] and [tex]v_{y}[/tex] denote the initial and final vertical velocity of the supply; [tex]u_{y} = 0\; {\rm m\cdot s^{-1}}[/tex] since the plane was flying horizontally.

Let [tex]x_{y}[/tex] denote the vertical displacement of the supply; [tex]x_{y} = 350\; {\rm m} - 810\; {\rm m} = (-460)\; {\rm m}[/tex].

Let [tex]a_{y}[/tex] denote the vertical acceleration of the supply; [tex]a = (-g) = (-9.8)\; {\rm m\cdot s^{-2}}[/tex].

Make use of the SUVAT equation [tex]{v_{y}}^{2} - {u_{y}}^{2} = 2\, a_{y}\, x_{y}[/tex] to find [tex]v_{y}[/tex], the final vertical velocity of the supply:

[tex]\begin{aligned} {v_{y}}^{2} &= {u_{y}}^{2} + 2\, a_{y}\, x_{y} \end{aligned}[/tex].

[tex]\begin{aligned} v_{y} &= -\sqrt{{u_{y}}^{2} + 2\, a_{y}\, x_{y}} \\ &= -\sqrt{0^{2} + 2\, (-9.8)\, (-460)}\; {\rm m\cdot s^{-1}} \\ &\approx (-94.953)\; {\rm m\cdot s^{-1}} \end{aligned}[/tex].

(Negative since the supply would be travelling downwards.)

Let [tex]t[/tex] denote time it takes for the supply to land on the hill after being dropped from the plane. Make use of the SUVAT equation [tex]t = (v_{y} - u_{y}) / (a)[/tex] to find the value of [tex]t\![/tex]:

[tex]\begin{aligned} t &= \frac{v_{y} - u_{y}}{a} \\ &\approx \frac{(-94.953) - 0}{(-9.8)} \; {\rm s}\\ &\approx 9.6890 \; {\rm s} \end{aligned}[/tex].

Apply unit conversion and ensure that [tex]v_{x}[/tex], the horizontal speed of the plane is in the standard unit [tex]{\rm m\cdot s^{-1}}[/tex]:

[tex]\begin{aligned} v_{x} &= \frac{260\; {\rm km}}{1\; {\rm h}} \times \frac{1\; {\rm h}}{3600\; {\rm s}} \times \frac{1000\; {\rm m}}{1\; {\rm km}} \\ &\approx 72.222\; {\rm m\cdot s^{-1}}\end{aligned}[/tex].

Under the assumptions, the horizontal speed of the supply will be the same as that of the plane- [tex]v_{x} \approx 72.222\; {\rm m\cdot s^{-1}}[/tex]- until it lands.

While in the air, the supply will travel a horizontal distance of:

[tex]\begin{aligned}x_{x} &= v_{x}\, t \\ &\approx 72.222\; {\rm m\cdot s^{-1}} \times 9.6890\; {\rm s} \\ &\approx 699.76\; {\rm m}\end{aligned}[/tex].

Hence, for the supply to land exactly at the top of the hill, the plane need to drop the supply while at a horizontal distance of approximately [tex]700\; {\rm m}[/tex] away from the hill.

The horizontal distance between the trees and the location where the plane dropped the supply would be approximately [tex](700\; {\rm m} - 30\; {\rm m}) = 670\; {\rm m}[/tex]. The time required for the the supply to reach that horizontal position would be:

[tex]\begin{aligned} t &= \frac{x_{x}}{v_{x}} \approx \frac{669.76\; {\rm m}}{72.222\; {\rm m\cdot s^{-1}}} \approx 9.2736\; {\rm s}\end{aligned}[/tex].

Let [tex]h_{0}[/tex] denote the initial height of the supply (relative to the sea level.) In this question, [tex]h_{0} = 810\; {\rm m}[/tex].

Let [tex]h(t)[/tex] denote the height of the supply (relative to the sea level) after being dropped from the plane for time [tex]t[/tex].

The SUVAT equation [tex]h(t) = (1/2)\, a\, t^{2} + u_{y}\, t + h_{0}[/tex] gives an expression for [tex]h(t)[/tex]. Make use of this equation to find the height of the supply (relative to the sea level) when the supply reach the horizontal position of the trees at [tex]t \approx 9.2736\; {\rm s}[/tex]:

[tex]\begin{aligned} h(t) &= \frac{1}{2}\, a\, t^{2} + u_{y}\, t + h_{0} \\ &= \frac{1}{2}\times (-9.8)\, (9.2736)^{2} + 0\times 9.2736 + 810 \\ &\approx 388.60\; {\rm m} \end{aligned}[/tex].

Note that the altitude of the top of the trees is [tex]350\; {\rm m} + 40\; {\rm m} = 390\; {\rm m}[/tex] relative to the sea level. Since [tex]388.90\; {\rm m} < 390\; {\rm m}[/tex], the supplies will run into the trees.

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