Speed And Velocity

Given, velocity,
As we know that,

Here, is the displacement of the particle.
Now,
Integrate on the both sides, limit to , we get




Average velocity,
Hence, the average velocity of the object between
to is

Step 1: Apply the principle of superposition of forces. Assuming linear elasticity, the change in length due to each force can be superimposed. Step 2: Calculate the original length . Using the relationship between force and elongation: Solving for with the known conditions gives:

Step 1: Applying the principle of conservation of momentum The force on the block due to the jet of water can be found using the principle of conservation of momentum. The rate of momentum transfer from the jet to the block is given by: Where: - is the mass flow rate of the water, and - is the velocity of the water. Substituting the given values: Step 2: Calculate the acceleration of the block The acceleration of the block is given by Newton's second law: Substituting the known values: Thus, the acceleration of the block is .

Step 1: Calculate the magnetic force needed to balance gravity. The gravitational force is given by: where (mass of the wire), (acceleration due to gravity). Step 2: Apply the formula for magnetic force. The magnetic force that balances the weight is given by: where is the current, (length of the wire), and (magnetic field strength). Setting , Rounded to two significant figures,

Step 1: Understanding the relation between work and force.
We know that work ( ) is defined as the product of force and distance moved in the direction of the force. Mathematically,
where:
is the work done,
is the force, and
is the distance moved.
Step 2: The dimensional formula for force.
Force is related to mass, length, and time by the second law of motion:
where:
is the mass,
is the length, and
is the time.
Step 3: Substituting the dimensional formula of force into the formula for work.
Now, substitute the dimensional formula of force into the equation for work:
Thus, the dimensional formula of work is:
Step 4: Solving for the dimensional formula of mass.
We are asked to find the dimensional formula for mass. From the above equation, we see that:
This means the dimensional formula for mass is:

Step 1: Calculating the Tension in the Wire Centripetal force provides tension: Step 2: Calculating Strain Strain is given by: Stress: Thus, the correct answer is option (1).

Given the equation of motion: To find the velocity , differentiate with respect to : Using the chain rule for as a function of , we get: This simplifies to: Now differentiate with respect to to find the acceleration (which is the derivative of velocity): Thus, the retardation, which is the negative acceleration, is:

We are given that the car passes a stationary observer while sounding a horn with a frequency of 1000 Hz. The ratio of frequencies before and after the car passes the observer is given as 11:9. The formula for the Doppler effect when the source is moving and the observer is stationary is: where: - is the frequency observed by the observer,
- is the frequency of the source (1000 Hz),
- is the speed of sound (340 m/s),
- is the speed of the source (the car’s speed, which we need to find).
Step 1: Set up the equation for frequencies
Before the car passes the observer, the frequency is , and after it passes, the frequency is . We are given the ratio of the frequencies before and after passing the observer: Substituting the Doppler shift equations for and , we get: Step 2: Solve for the speed of the car
Simplifying the equation: Now, cross-multiply to solve for : Expanding both sides: Substitute : Thus, the speed of the car is .

The velocity of the particle can be found using the work-energy theorem, which relates the change in velocity to the area under the acceleration versus position curve.
Given that the velocity at is , and from the graph, the area under the curve between and is the work done,
which contributes to the change in velocity. From the graph, we can estimate the area, which leads to a velocity of at .
Thus, the velocity at is:

Step 1: Using Energy Conservation The total mechanical energy of the particle is: For escape velocity: Step 2: Applying Given Condition The initial velocity given is : Substituting : At infinity, kinetic energy remains: Solving for : Thus, the correct answer is option (3).

Step 1: Using Stoke’s Law The velocity before entering water: Solving, we get: Thus, the correct answer is option (3).

The kinetic energy acquired by the particle is equal to the work done on it by the electric field, which is given by the change in electric potential energy: where C (charge of the particle), and V (potential difference). Calculating the kinetic energy: Using the kinetic energy formula , where kg (mass of the particle), we solve for (velocity):

To find the average angular velocity, we need to consider the total angular displacement over the total time taken. For a complete revolution in a circle, the total angular displacement is radians. Given that the particle completes half the circumference in 4 seconds and the other half in 2 seconds, the total time for one complete revolution is: Thus, the average angular velocity is given by: