To determine the changes in momentum and kinetic energy when velocity is doubled, we will use their respective formulas. Momentum is directly proportional to velocity, while kinetic energy is proportional to the square of velocity.
Define initial momentum and kinetic energy:
Let the mass of the object be \(m\) and its initial velocity be \(v\).
The initial momentum (\(P_1\)) is given by:
\[ P_1 = m \times v \]The initial kinetic energy (\(KE_1\)) is given by:
\[ KE_1 = \frac{1}{2} m v^2 \]Calculate new momentum and kinetic energy when velocity is doubled:
If the velocity is doubled, the new velocity (\(v_2\)) will be \(2v\).
The new momentum (\(P_2\)) will be:
\[ P_2 = m \times (2v) = 2mv \]Comparing \(P_2\) with \(P_1\):
\[ P_2 = 2 \times (mv) = 2 P_1 \]So, the momentum increases by 2 times.
The new kinetic energy (\(KE_2\)) will be:
\[ KE_2 = \frac{1}{2} m (2v)^2 \]\[ KE_2 = \frac{1}{2} m (4v^2) \]\[ KE_2 = 4 \times \left(\frac{1}{2} m v^2\right) \]Comparing \(KE_2\) with \(KE_1\):
\[ KE_2 = 4 KE_1 \]So, the kinetic energy increases by 4 times.
Conclusion:
When the velocity is doubled, the momentum increases by 2 times, and the kinetic energy increases by 4 times.
C) Momentum increases by 2 times and kinetic energy by 4 times. This matches our calculations.
In each of the questions below is given a statement followed by two assumptions I and II. You have to consider the statement and the following assumptions and decide which of the assumptions are implicit in the statement.
Statements:
Everyone loves to read fairy tales.
Assumptions:
I. Fairy tales are only reading material.
II. Nobody likes other stuff.
Only assumption I is implicit.
Both I and II are implicit
Neither I nor II is implicit
Only Assumption II is implicit.
This question tests logical reasoning, specifically the ability to identify implicit assumptions. An assumption is something taken for granted or supposed without proof. For an assumption to be implicit, it must be a necessary underlying belief for the statement to hold true. We need to evaluate if the given assumptions are absolutely essential for the statement to make sense.
Correct Option: C) Neither I nor II is implicit
The question asks to identify the agricultural product that generates the maximum export value among the given options. This requires knowledge of India's agricultural export statistics.
D) Coffee: Coffee, especially varieties like Indian Robusta and Arabica, fetches good prices in the international market. India is the sixth-largest producer and fifth-largest exporter of coffee in the world. The high unit value and consistent demand for Indian coffee contribute to its significant export earnings, often surpassing other traditional agricultural exports like spices, cotton, and jute in terms of overall value.
This question tests general knowledge about film awards, specifically the Jio Filmfare Awards 2018. To answer correctly, one needs to recall the winners of the major categories for that particular year.
The 63rd Filmfare Awards ceremony, honoring the best of Hindi cinema in 2017, was held on January 20, 2018. Vidya Balan won the Best Actress in a Leading Role award for her performance in the film 'Tumhari Sulu'.
C) Vidya Balan won the Best Actress in a Leading Role award at the Jio Filmfare Awards 2018 for her performance in the film 'Tumhari Sulu'.
First, we need to find the principal amount using the given simple interest for one year. Then, we will calculate the compound interest for the same principal, rate, and period, but with interest compounded half-yearly.
Given simple interest (SI) for 1 year = Rs. 400.
Rate of interest (R) = 10% per annum.
Time (T) = 1 year.
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]Substitute the given values:
\[ 400 = \frac{P \times 10 \times 1}{100} \] \[ 400 = \frac{10P}{100} \] \[ 400 = \frac{P}{10} \] \[ P = 400 \times 10 \] \[ P = \text{Rs. } 4000 \]Principal (P) = Rs. 4000
Annual rate (R) = 10%
Time (T) = 1 year
When interest is compounded half-yearly, the rate is halved, and the time period is doubled.
New rate per half-year \((r)\) = \[\frac{R}{2} = \frac{10\%}{2} = 5\%\]
Number of compounding periods \((n)\) = \(T \times 2 = 1 \times 2 = 2\) periods
The formula for the amount (A) with compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]Substitute the values:
\[ A = 4000 \left(1 + \frac{5}{100}\right)^2 \] \[ A = 4000 \left(1 + 0.05\right)^2 \] \[ A = 4000 \left(1.05\right)^2 \] \[ A = 4000 \times 1.1025 \] \[ A = \text{Rs. } 4410 \]Now, calculate the compound interest (CI):
\[ CI = A - P \] \[ CI = 4410 - 4000 \] \[ CI = \text{Rs. } 410 \]Correct Option: C) Rs. 410
If a = 0.125, then what is the value of \( \sqrt{4a^2-4a+1} + 3a \)?
The problem asks us to evaluate an algebraic expression involving a square root and a given value for the variable 'a'. The key to solving this is to simplify the expression inside the square root first, recognizing it as a perfect square trinomial, and then substitute the given value of 'a'.
We need to find the value of \( \sqrt{4a^2-4a+1} + 3a \).
The expression inside the square root is \( 4a^2-4a+1 \). This is a perfect square trinomial of the form \( (x-y)^2 = x^2 - 2xy + y^2 \).
Here, \( x^2 = 4a^2 \Rightarrow x = 2a \).
And \( y^2 = 1 \Rightarrow y = 1 \).
Check the middle term: \( -2xy = -2(2a)(1) = -4a \). This matches the given expression.
So, \( 4a^2-4a+1 = (2a-1)^2 \).
The expression becomes \( \sqrt{(2a-1)^2} + 3a \).
We know that \( \sqrt{x^2} = |x| \). Therefore, \( \sqrt{(2a-1)^2} = |2a-1| \).
So, the expression is \( |2a-1| + 3a \).
Given \( a = 0.125 \).
First, calculate \( 2a-1 \):
\[ 2a-1 = 2(0.125) - 1 \]\[ 2a-1 = 0.250 - 1 \]\[ 2a-1 = -0.750 \]Since \( 2a-1 = -0.750 \), which is negative, \( |2a-1| = |-0.750| = 0.750 \).
Substitute this back into \( |2a-1| + 3a \):
\[ 0.750 + 3(0.125) \]\[ 0.750 + 0.375 \]\[ 1.125 \]Correct Option: B) 1.125
To solve this problem, we need to identify the letters corresponding to the given positions in the English alphabet. Then, we will arrange these letters to form a meaningful word and finally identify its second letter.
The letters are A, S, T, Y.
Correct Option: A) T is the second letter of the meaningful word "STAY" formed by the given letters.
The question asks to identify an animal whose heart has only two chambers. This requires knowledge of the circulatory systems and heart structures of different animal groups, specifically focusing on vertebrates.
D) Rohu: Rohu is a type of freshwater fish. Fish belong to the class Pisces, and all fish have a two-chambered heart (one atrium and one ventricle) for single circulation.
According to the World Bank's Global Economic Prospects Report 2018, India may top the list as the fastest growing economy with a growth rate of ________.
The question asks for India's projected growth rate according to the World Bank's Global Economic Prospects Report 2018. This requires recalling specific economic data from that particular report.
B) 7.30% is the correct growth rate projected for India by the World Bank's Global Economic Prospects Report 2018, making it the fastest-growing economy among major nations.