A 227 kg motorcycle moves with a velocity of 8 m/s. What is its kinetic energy? ______

(a) Consider the initial-value problem dA/dt = kA, A(0) = A0 as the model for the decay of a radioactive substance. Show that, i

murzikaleks [220]

Answer:

a) $t = -\frac{ln(2)}{k}$

b) See the proof below

$A(t) = A_o 2^{-\frac{t}{T}}$

c) $t = 3T \frac{ln(2)}{ln(2)}= 3T$

Explanation:

Part a

For this case we have the following differential equation:

$\frac{dA}{dt}= kA$

With the initial condition $A(0) = A_o$

We can rewrite the differential equation like this:

$\frac{dA}{A} =k dt$

And if we integrate both sides we got:

$ln |A|= kt + c_1$

Where $c_1$ is a constant. If we apply exponential for both sides we got:

$A = e^{kt} e^c = C e^{kt}$

Using the initial condition $A(0) = A_o$ we got:

$A_o = C$

So then our solution for the differential equation is given by:

$A(t) = A_o e^{kt}$

For the half life we know that we need to find the value of t for where we have $A(t) = \frac{1}{2} A_o$ if we use this condition we have:

$\frac{1}{2} A_o = A_o e^{kt}$

$\frac{1}{2} = e^{kt}$

Applying natural log we have this:

$ln (\frac{1}{2}) = kt$

And then the value of t would be:

$t = \frac{ln (1/2)}{k}$

And using the fact that $ln(1/2) = -ln(2)$ we have this:

$t = -\frac{ln(2)}{k}$

Part b

For this case we need to show that the solution on part a can be written as:

$A(t) = A_o 2^{-t/T}$

For this case we have the following model:

$A(t) = A_o e^{kt}$

If we replace the value of k obtained from part a we got:

$k = -\frac{ln(2)}{T}$

$A(t) = A_o e^{-\frac{ln(2)}{T} t}$

And we can rewrite this expression like this:

$A(t) = A_o e^{ln(2) (-\frac{t}{T})}$

And we can cancel the exponential with the natural log and we have this:

$A(t) = A_o 2^{-\frac{t}{T}}$

Part c

For this case we want to find the value of t when we have remaining $\frac{A_o}{8}$

So we can use the following equation:

$\frac{A_o}{8}= A_o 2^{-\frac{t}{T}}$

Simplifying we got:

$\frac{1}{8} = 2^{-\frac{t}{T}}$

We can apply natural log on both sides and we got:

$ln(\frac{1}{8}) = -\frac{t}{T} ln(2)$

And if we solve for t we got:

$t = T \frac{ln(8)}{ln(2)}$

We can rewrite this expression like this:

$t = T \frac{ln(2^3)}{ln(2)}$

Using properties of natural logs we got:

$t = 3T \frac{ln(2)}{ln(2)}= 3T$

8 0

3 years ago

Parents have a major role in observing how young children grow at home. Create and discuss simple developmental monitoring tips

Rudik [331]

Answer:

by observation of their children

4 0

2 years ago

A railroad freight car of mass 3.34 × 104 kg collides with a stationary caboose car. They couple together, and 19.0% of the init

Tresset [83]

Answer:

4.123 * 10∧4 kg

Explanation:

mass of car M' = 3.34 * 10∧4 kg

energy loss E = 19/100 K

from law of conservation of momentum ,

M' V' = ( M' + M'' ) v

V = M' V' / ( M' + M'' )

Initial kinetic energy is K' = (M' V')² / 2

final kinetic energy K" = 81 K' /100

= (M' V')² 81 / (200) = ( M' + M'' ) v²/ 2

therefore , M' / ( M' + M" ) = 0.81

mass of caboose is , M" = 1.234 M' - M'

M" = .234 M'

= 0,234 ( 3.34 * 10∧4 kg)

= 4.123 * 10∧4 kg

8 0

2 years ago

A dandelion seed floats to the ground in a mild wind with a resultant velocity of 26.0 cm/s. If the horizontal component velocit

stira [4]

Answer:

24 cm/s

Explanation:

Applying

Pythagoras theorem,

a² = b²+c²............. Equation 1

Where a = resultant, b = vertical component, c = horizontal component

From the question,

Given: a = 26 cm/s, c = 10 cm/s

Substitute these values into equation 1

26² = b²+10²

676 = b²+100

b² = 676-100

b² = 576

b = √576

b = 24 cm/s

7 0

2 years ago

Which physicist first identified the electron as an independent part of the atom?

Simora [160]

<h2>Answer: J. J. Thomson </h2>

Thomson conducted a series of experiments on cathode ray tubes, which led him to the discovery of electrons.

This is how on April 30th, 1897 Thomson announced the discovery of the electron (although at that time he called it a corpuscle)

This discovery was considered one of the milestones in science of the late nineteenth century, since it allowed to have a new conception of the structure of matter and its interaction with energy.

It should be noted that from this discovery, Thomson proposed a new model of the atom that although erroneous, remained valid for years and served as the basis for new postulates and research on the subject.

7 0

3 years ago

A 227 kg motorcycle moves with a velocity of 8 m/s. What is its kinetic energy? _______ Joules