Motion In A Straight Line

Step 1: Concept
Kepler's Third Law: .
Step 2: Analysis
. Given , then . .
Step 3: Conclusion
Therefore, .
Final Answer: (B)

Step 1: Concept
According to Kepler's Third Law, .
Step 2: Analysis
. Given and hours. .
Step 3: Conclusion
The new time period hours.
Final Answer: (C)

Step 1: Concept
Time period .
Step 2: Analysis
New length . New time period .
Step 3: Conclusion
Percentage increase .
Final Answer: (B)

Step 1: Formula
The root mean square velocity is .
Step 2: Substitution
.
Step 3: Calculation
.
Final Answer: (a)

Step 1: Definition
Condensation polymers are formed by the combination of monomers with the elimination of small molecules like water, methyl alcohol, etc.
Step 2: Analysis
- Rubber, PVC, and polyethene are addition polymers. - Proteins are natural polymers formed by the condensation of -amino acids.
Step 3: Conclusion
During the formation of proteins, amino acids link together via peptide bonds ( ) with the release of water molecules.
Final Answer: (b)

Step 1: Definition of Specific Conductance
Specific conductance ( ) is the conductance of 1 cc of solution.
Step 2: Effect of Dilution
Upon dilution, the total volume increases, which means the number of ions per cc (unit volume) decreases.
Step 3: Conclusion
Because there are fewer ions available to carry current in a given unit volume, the specific conductance decreases with dilution.
Final Answer: (c)

Step 1: Values
.
Step 2: Equation
.
Step 3: Calculation
. .
Final Answer: (C)

Step 1: Acceleration in SHM.
In simple harmonic motion (SHM), the acceleration is related to the displacement from the mean position by the equation: where is the angular frequency and is the displacement.

Step 2: Use the given data.
We are given:
-
- Substitute these values into the SHM equation:

Step 3: Solve for angular frequency .
Rearranging the equation:

Step 4: Calculate the time period.
The time period of the particle is related to the angular frequency by the formula: Substitute : Thus, the time period is approximately .

Step 5: Conclusion.
Therefore, the time period is .

Step 1: Formula for Escape Velocity.
The escape velocity is given by the formula: where: - is the gravitational constant, - is the mass of the planet, and - is the radius of the planet.

Step 2: Understand the Given Data.
We are told that: - The radius of the new planet is double that of Earth, so . - The density of the new planet is the same as Earth, meaning the mass of the new planet will be 8 times the mass of Earth. This is because the mass of a planet depends on its volume, and volume scales with the cube of the radius. Therefore, if the radius is doubled, the volume (and thus the mass) increases by a factor of .

Step 3: Relation for New Planet.
The escape velocity depends on both the mass and the radius of the planet. The formula for the escape velocity can be written for the new planet as: Substitute and into the equation:

Step 4: Simplifying the Expression.
Now simplify the expression:

Step 5: Compare with Escape Velocity from Earth.
We know that the escape velocity from Earth is: Thus, we can write:

Step 6: Conclusion.
Therefore, the escape velocity from the new planet is .

Step 1: Understanding the Concept:
Use parallel axis theorem: . For disc about diameter, .
Step 2: Detailed Explanation:
Axis is tangential to circumference and parallel to diameter. Distance from centre to tangent = .
.
Step 3: Final Answer:
Thus, .

Step 1: Understanding the Concept:
Escape velocity .
Step 2: Detailed Explanation:
For earth: .
For planet: , .
.
Step 3: Final Answer:
Thus, escape velocity = .

Concept:

Step 1: Analyze acceleration graph.
Acceleration increases linearly with time, then becomes zero.

Step 2: Effect on velocity.

• Increasing acceleration velocity curve is concave upward (curved increase)
• Constant zero acceleration velocity becomes constant (horizontal line)

Step 3: Match with options.
Only option (C) shows:
• Initially curved increase
• Then constant velocity Conclusion:

Step 1: Find position.


Step 2: Use .
Let :

Step 3: Find acceleration.
Conclusion:

Step 1: Understanding the Concept:
Time of flight for one ball: . Balls are thrown at regular intervals of 1 second.
Step 2: Detailed Explanation:
At , one ball is thrown; previous balls have been in air for 1 s, 2 s, and 3 s. Height formula: . For : . For : . For : . Heights are: 15 m, 20 m, 15 m.
Step 3: Final Answer:

Concept: Acceleration is the second derivative of displacement:

Step 1: Given equation from graph.
From the graph:

Step 2: Find velocity.


Step 3: Find acceleration.


Step 4: Conclusion.
Acceleration is constant with time.

Concept: Centripetal force: Also:

Step 1: Force relation.


Step 2: Substitute velocity.


Step 3: Solve relation.

Concept: Acceleration is the derivative of velocity: Position is obtained by integrating velocity.

Step 1: Find velocity.
Since initially at rest at :

Step 2: Find displacement.
At ,

Step 3: Distance in 2 s.

Concept: The argument of a trigonometric function must be dimensionless. Hence, Also, displacement has dimension of length.

Step 1: Dimension of .
Since , and sine is dimensionless:

Step 2: Dimension of .
From dimensionless:

Step 3: Dimension of and .
From and dimensionless:

Step 4: Dimension of .
Since options match closest to frequency-type dimension:

Concept: Escape velocity: For same density:

Step 1: Relate mass and radius.


Step 2: Use escape velocity formula.


Step 3: Final relation.

Concept: For small heights ,

Step 1: Apply condition

Step 2: Simplify Conclusion

Concept: Maximum height of projectile:

Step 1: Heights of two projectiles:

Step 2: Using identity:

Step 3: Ratio:

Concept: Relative motion:

Step 1:Max distance when velocity difference = 0}

Step 2:Substitute:}