Aman tosses a dart upward with a velocity of 14.1 m/s at 60° angle. How much time is it aloft?
What is its max height and range?

Angle of projection of the dart = 60°

initial velocity of the dart = 14.1 m/s

Using basic trigonometry, we can see that:

Vertical component of the Dart = 14.1*Sin 60° = 12.2 m/s

Horizontal component of the Dart = 14.1*Cos 60° = 7.05 m/s

Explaining the concept:

Once the dart is in mid-air, there will be no force that will reduce its horizontal velocity

So, the dart will be moving at a constant horizontal velocity of 7.05 m/s

but the force of gravity will be applied on the dart while mid-air and will pull it downwards with an acceleration of 9.8 m/s²

Hence, the time the dart will be aloft is the same as the time taken by the dart to reach the ground if we threw it vertically upwards at a velocity of 12.2 m/s

Solving for the time taken for the dart to reach the ground:

initial velocity = 12.2 m/s          

velocity at max height = 0 m/s      

acceleration = -9.8 m/s

time taken to reach max height:

v = u + at            using the first equation of motion

replacing the variables

0  = 12.2 + (-9.8)t

t = -12.2 / -9.8

t = 1.24 seconds

Time taken to reach the ground:

time taken to reach the ground = 2 * time taken to reach max height

(Since the dart will take the same time to come back down)

Time taken to reach the ground = 2 * 1.24

Time taken to reach the ground = 2.48 seconds

Therefore, the dart will hit the ground and stop 2.48 seconds after throwing

Explaining the concept:

From newton's second equation of motion

s = ut + 1/2 at²

Since we need the max vertical height, we need the vertical component of the values,

so this equation can be rewritten for vertical height as :

s(vertical) = u(vertical)* t + 1/2 * a(vertical) * t²

from the last section, we know that the dart reaches its maximum height at  t = 1.24 seconds

replacing the values in this equation:

s(vertical) = (12.2 * 1.24) + 1/2 * (-9.8) * (1.24)²

s(vertical) = 15.128 + (-7.5)

s(vertical) = 7.63 m

Therefore, the maximum height of the dart is 7.63 m

Explaining the concept:

The range of the dart is the horizontal distance covered by the dart

from the second section, we know that the dart travels horizontally with a constant velocity of 7.05 m/s

from the third section, we know that the total time taken by the dart to hit the ground is 2.48 seconds

Since the horizontal velocity of the dart is constant, we can say that it moved horizontally for 2.48 seconds at a constant velocity of 7.05 m/s

Solving for the range:

s = ut + 1/2 at²            From the second equation of motion

this equation can be rewritten for horizontal distance as:

s(horizontal) = u(horizontal)* t + 1/2 (a)(t)²

The dart is moving at a constant velocity, that means that its acceleration is 0

s(horizontal) = u(horizontal)* t + 1/2 (0)(t)²

s(horizontal) = u(horizontal)* t

replacing the variables

s(horizontal) = 7.05 * 2.48   [here, 2.48 is the time taken by the dart to reach the ground and stop]

s(horizontal) = 17.484 m

The horizontal distance covered by the dart is 17.484 m

Therefore, the range of the dart is 17.484 m

time =  2.49 s

max height = 7.61 m

range = 17.57 m

See below for exact values.

Explanation:

Since we are given the angle at which the object is tossed, this object is in projectile motion.

We are given the initial velocity of the dart, and we can use the angle to solve for its horizontal and vertical components.

Horizontal component:

Vertical component:

In order to find the time that the dart stays in the air, we can use the constant acceleration equation that does not include displacement. This equation is:

Since we solve for time using the vertical motion of the projectile, we will use this equation in terms of the y-direction.

We don't know what the final velocity of the dart when it reaches the ground is, but we do know that its final velocity when it reaches its maximum height is 0 m/s.

Therefore, we can solve for half of the full time that it takes the object to reach the ground, and double this value at the end.

Let's set the downwards direction to be negative and the upwards direction to be positive.

Using our knowledge and previous calculations, we know that:

The acceleration of gravity is in the y-direction and facing downwards, so that's why it is -9.8 m/s².

Substitute these values into the constant acceleration equation.

Subtract 14.1 * sin(60) from both sides of the equation.

Divide both sides of the equation by -9.8 to solve for t.

Now, this is only the time for half of the object's trajectory. Double it to find the total time the dart is aloft:

The dart is in the air for a total of 2.49 seconds.

We can find the maximum height of the dart by using another constant acceleration equation.

Since we don't have the final velocity of the object, we can use this equation:

Subtract from both sides to get the change in position, or delta x.

In order to find the maximum height, we need to use this equation in terms of the y-direction.

Remember that time is the same regardless of the x- or y- direction.

Now, we can solve for the displacement in the y-direction by plugging in the values that we know.

Note that we are using half of the time t, since this is where the maximum height occurs.

Plug these known values into the constant acceleration equation:

The maximum height of the dart is 7.61 meters.

Finding the range of the object in projectile motion involves the same constant acceleration equation, but this time we are solving for the displacement in the horizontal direction (x-direction). Therefore:

We know the horizontal component of the initial velocity vector, which is what we will be using.

The acceleration, of an object in projectile motion, in the x-direction is always 0 m/s².

We will use the full time of the object since we want to find the entire horizontal distance that the object covers. We have:

Plug these values into the constant acceleration equation.

The range of the dart is 17.57 meters.

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.

scattering of light by tiny particles in air to produce the colors we see on sky, is dependent on the following factors:

(i) wavelength of light

(ii) size of the scattering particle  

(iii) distance traveled by light through the atmosphere.

according to rayleigh's law of scattering, the amount of scattering is inversely proportional to the fourth power of its wavelength. shorter the wavelength, greater is its scattering. hence violet/blue light is scattered more than red.

sun's light is a composite light ( visible light) with wavelengths in the range 400 nm (violet) to 700 nm (red). as sunlight passes through our atmosphere, the molecules of air scatter light according to their wavelengths. during sunrise and sunset, the sun's position is below the horizon. hence light from the sun travels a greater distance in the atmosphere when compared to its position a noon.

as sunlight enters the earth's atmosphere, blue light is scattered away and at the end of its journey in the sky, only red and orange wavelengths remain in the beam.

hence at sunrise and sunset, the sky appears reddish orange.