For an object undergoing constant acceleration, the equation v = v₀ + at shows that the final velocity is obtained by adding the product of acceleration and time to the initial velocity. This linear relationship is fundamental in kinematics.

Newton's First Law, often called the law of inertia, states that without a net external force, an object will maintain its state of rest or constant velocity. This principle is key in understanding motion.

When a spring is stretched or compressed, it stores energy in the form of elastic potential energy. This energy can be released to do work when the spring returns to its equilibrium position.

In an isolated system with no net external forces, momentum is conserved even if other quantities like kinetic energy may transform. This conservation principle underpins the analysis of collisions.

A resistor is designed to hinder the flow of electric current by converting electrical energy into heat, following Ohm's Law. This opposition to current flow is fundamental in circuit design.

Using the formula v = gt for an object in free fall, where g ≈ 9.8 m/s² and t = 3 s, the speed calculates to v = 9.8 × 3 = 29.4 m/s. This is a basic application of kinematics.

When friction slows an object, it does work opposite to the direction of motion, meaning the work done is negative. This negative work decreases the object's kinetic energy.

In projectile motion ignoring air resistance, the horizontal velocity is constant because no horizontal forces act on the ball. Meanwhile, gravity causes constant vertical acceleration.

Initial kinetic energy is 0.5×2×(5²) = 25 Joules and the final kinetic energy is 0.5×2×(15²) = 225 Joules. The change in kinetic energy is 225 - 25 = 200 Joules.

On an inclined plane, the component of gravitational force acting parallel to the surface is given by mg sin(θ). This force is what drives the object to slide down.

In inelastic collisions, where objects stick together, momentum is conserved even though kinetic energy may not be. This principle is essential in analyzing collision problems.

In a series circuit, resistors are connected end-to-end and the total resistance is simply the sum of all individual resistances. This linear addition is a fundamental property of series circuits.

In simple harmonic motion, the restoring force (and therefore acceleration) is proportional to the displacement. Thus, at the maximum displacement points, the acceleration reaches its maximum magnitude, acting in the opposite direction of the displacement.

For small angular displacements, the approximation sin(θ) ≈ θ holds, making the period of a simple pendulum dependent only on its length and gravitational acceleration, not the amplitude. This is why the period remains nearly constant for small swings.

At the bottom of the circle, the tension must provide the centripetal force and counteract gravity, resulting in a higher tension. At the top, gravity assists in providing the centripetal force, thereby reducing the tension required in the string.

The period of a mass-spring system is given by T = 2π√(m/k). Doubling the mass changes the period to T' = 2π√(2m/k), which is √2 times the original period. This relation highlights how mass influences oscillatory motion.

First, apply Newton's second law to find the acceleration: a = F/m = 20/5 = 4 m/s². Then use the displacement formula d = 0.5at² = 0.5×4×(4²) = 32 meters.

Using conservation of momentum, the total momentum before collision is (3 kg × 8 m/s) = 24 kg·m/s. After collision, the combined mass is 5 kg, so the common velocity is 24/5 = 4.8 m/s.

Centripetal acceleration is given by a = v²/r. When the radius is halved and the speed remains constant, the acceleration becomes a = v²/(r/2) = 2(v²/r), meaning it doubles.

In an RC circuit, the capacitor charges following an exponential behavior. It typically reaches approximately 63% of its final voltage after one time constant (τ = RC), which is a hallmark of capacitor charging dynamics.