Newtons Laws Of Motion

Step 1: For a given range with the same initial speed , a projectile can be fired at two complementary angles and .
Step 2: Time of flight for a projectile is: Hence,
Step 3: Find the product :
Step 4: The range of a projectile is:
Step 5: Substitute :

Step 1: Identify the forces acting on the block. Upward force = Normal reaction Downward force = Weight
Step 2: Apply Newton’s second law to the block (upward positive):
Step 3: Solve for the normal reaction:
Step 4: Find the displacement of the block in time . Since the platform starts from rest with acceleration :
Step 5: Calculate the work done by the normal reaction:

Step 1: Let the length of the rod be
Step 2: The centre of mass of the rod is at a distance from the pivot. The vertical fall of the centre of mass when the rod moves from to the horizontal is:
Step 3: Loss of potential energy:
Step 4: Gain in rotational kinetic energy: For a rod about one end,
Step 5: Apply conservation of energy:
Step 6: Simplify:
Step 7: Substitute : Taking ,

Step 1: Find the radius of the disc. Given diameter ,
Step 2: The rod forms a tangent in the plane of the disc. Hence, the axis is parallel to a diameter of the disc and is at a distance from the centre.
Step 3: Moment of inertia of a thin disc about a diameter:
Step 4: Use the parallel axis theorem to find the moment of inertia about the tangent:
Step 5: Radius of gyration is defined by:
Step 6: Substitute m:

Step 1: Express the gravitational constant in terms of and .
Step 2: Identify the initial and final distances of the moon’s centre from the centre of the earth. Since the moon has radius , it strikes the earth when the distance between centres is:
Step 3: Use conservation of mechanical energy. Initial velocity is zero, so:
Step 4: Substitute the values:
Step 5: Simplify:
Step 6: Hence,

Step 1: Given data:
Step 2: Using the equation of motion, Negative sign indicates acceleration is upward with magnitude .
Step 3: Applying Newton’s second law (upward positive):
Step 4: Substituting values, Hence, the tension in the cable is

Step 1: The system is an Atwood machine with masses Acceleration of the system:
Step 2: Velocity of the 2 kg mass after 5 s:
Step 3: After the string breaks, the 2 kg mass moves upward with initial velocity against gravity. Maximum additional height reached:
Step 4: Hence, from the moment the string breaks, the 2 kg mass rises by

Step 1: Recall the nature of an LC circuit. When a charged capacitor is connected to an inductor, the system performs electromagnetic oscillations with angular frequency:
Step 2: Write expressions for energies. Electric energy stored in the capacitor at time : Magnetic energy stored in the inductor:
Step 3: Express charge and current as functions of time. For an LC circuit:
Step 4: Substitute into energy expressions. Electric energy: Magnetic energy: Since ,
Step 5: Condition for equal energies.
Step 6: Find the required time.
Hence, the correct answer is .

Step 1: To cross the river, only the component of velocity perpendicular to the bank is useful.
Step 2: Effective crossing speed:
Step 3: Time to cross width:
Step 4: Flow velocity affects drift, not crossing time. Hence → (C).

Step 1: Resultant force on system:
Step 2: Mass of system kg. Acceleration components:
Step 3: Angle with axis:
Step 4: Hence → (C).

Step 1: From the velocity–time graph, determine acceleration at s. Between s and s the graph shows velocity decreasing linearly from m/s to m/s.
Step 2: Acceleration in this interval is constant:
Step 3: For a lift moving upward, tension is given by: Take m/s .
Step 4: But the graph in figure used standard m/s in options approximation. Using :
Step 5: Closest option → (D).

Step 1: In elastic collision of identical masses, apply conservation of kinetic energy: where is speed of second mass.
Step 2:
Step 3: Apply conservation of momentum vectorially. If initial momentum is along axis and first mass goes along axis after collision, components must satisfy: Magnitude of momentum of second mass becomes:
Step 4: For identical masses in elastic collision, final velocities are perpendicular only when magnitudes are equal to initial . Checking options, only consistent natural result is . Hence → (C).

Step 1: Recall the formula for gravitational potential energy. Step 2: Substitute the given values. Step 3: Interpret the result. Since the stone is raised, its gravitational potential energy increases. Step 4: Final conclusion. The stone gains gravitational potential energy of joules.

Step 1: Understand how friction arises. Friction is caused by the interlocking of irregularities between two surfaces in contact. Step 2: Analyze each method given.

Using a smooth object: reduces surface irregularities → friction decreases
Using a smooth plane: smoother contact → less interlocking
Providing a lubricant: creates a thin layer between surfaces, reducing direct contact
Step 3: Evaluate the options. All the given methods help in reducing friction. Step 4: Final conclusion. The effect of frictional force may be minimized by all of these methods.

Step 1: Recall the classification of physical quantities. Physical quantities are classified into:

Fundamental (base) quantities
Derived quantities
Step 2: Identify the nature of temperature. Temperature is one of the seven fundamental (base) quantities in the SI system. It is independent and is not derived from length (L), mass (M), or time (T). Step 3: Check the given options.

(A) Length and mass — incorrect
(B) Mass and time — incorrect
(C) Length, mass and time — incorrect
(D) None of these — correct ✔
Step 4: Final conclusion. Since temperature is a fundamental quantity, it cannot be expressed as a derived quantity in terms of L, M, and T.

Step 1: Recall the definition of electron volt. An electron volt (eV) is defined as the amount of energy gained by an electron when it is accelerated through a potential difference of . Step 2: Express electron volt mathematically. Step 3: Analyze the options.

(A) Potential difference — measured in volts
(B) Charge — measured in coulombs
(C) Energy — correct ✔
(D) Capacity — measured in farads
Step 4: Final conclusion. Electron volt is a unit of .

Step 1: Recall the concepts of speed and velocity.

Speed is a scalar quantity (magnitude only).
Velocity is a vector quantity (magnitude and direction).
Step 2: Analyze motion in a circular path. When a body moves in a circular path at constant speed:

The magnitude of velocity remains constant.
The direction of velocity keeps changing at every point.
Step 3: Understand acceleration in circular motion. There is always a centripetal acceleration directed towards the centre of the circle.

Its magnitude remains constant (if speed and radius are constant).
Its direction keeps changing as the body moves along the circle.
Step 4: Evaluate the options.

(A) Incorrect — velocity is not constant
(B) Correct — both velocity and acceleration change direction ✔
(C) Incorrect — neither increases
(D) Incorrect — velocity direction also changes
Step 5: Final conclusion. The correct answer is .