The equation s = ut + (1/2)at^2 is the standard kinematic formula for displacement under constant acceleration. It accurately accounts for both the object's initial velocity and the acceleration over time.

Newton's Second Law states that the net force acting on an object is equal to its mass multiplied by its acceleration. This relationship is succinctly expressed as F = ma.

Kinetic energy represents the energy an object possesses due to its motion. The formula KE = (1/2)mv^2 correctly calculates this energy using the object's mass and the square of its velocity.

Force in the International System is measured in Newtons. One Newton is defined as the force required to accelerate a 1 kg mass at 1 m/s².

An object in free fall is predominantly influenced by gravitational force, which pulls it towards the Earth. Other forces like friction or tension are negligible or not present in free-fall conditions.

Acceleration is the change in velocity divided by the time required for that change. Here, 20 m/s divided by 5 seconds yields an acceleration of 4 m/s².

The horizontal component of the velocity is computed as the initial speed multiplied by the cosine of the launch angle. With cos(60°) equal to 0.5, the horizontal component is 30 m/s × 0.5, which equals 15 m/s.

In collisions where external forces are negligible, the total momentum of the system remains constant. This makes the conservation of momentum the key principle for analyzing such events.

Even though the speed remains constant, an object in circular motion is continuously changing direction, resulting in centripetal acceleration toward the center of the circle.

At the peak of a hill, the roller coaster slows down as it gains height, meaning kinetic energy is transformed into gravitational potential energy. In a frictionless system, total mechanical energy is conserved.

Using Newton's second law, a = F/m. Dividing the frictional force of 4 N by the mass of 2 kg results in a deceleration of 2 m/s².

The moment of inertia for a rod about an axis through one end is derived to be (1/3)mL². This result comes from integrating the contributions of each infinitesimal mass element along the rod's length.

At its maximum displacement, the pendulum momentarily comes to rest before changing direction, which means its speed is zero and its gravitational potential energy is at a maximum.

Friction is a non-conservative force because it dissipates mechanical energy as heat, making the work done path-dependent. This contrasts with conservative forces like gravity, which conserve mechanical energy.

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. This fundamental principle links the concepts of work and energy in physics.

A rolling disk possesses both translational and rotational kinetic energy. Using its moment of inertia (1/2)MR² and the no-slip condition (v = ωR), the total kinetic energy combines to (1/2)Mv² + (1/4)Mv² = (3/4)Mv².

The work done by a variable force is found by integrating the force with respect to displacement. Integrating kx³ from 0 to x results in (k/4)x❴, showing a fourth-power dependence on x.

For an Atwood machine, the net acceleration is determined by the difference between the two masses divided by their sum, multiplied by g. This gives the formula a = ((m₝ - m₂) / (m₝ + m₂))g.

By decomposing the initial velocity into horizontal and vertical components and eliminating time, the range of a projectile in a vacuum is derived as R = (v₀² sin(2θ)) / g.

Newton's second law for rotation states that the net torque on a body equals the product of its moment of inertia and its angular acceleration (τ = Iα). Therefore, the angular acceleration is given by α = τ / I.