Kinematics Formula – Definition, Solved Examples

Kinematics, a fundamental branch of physics, studies how objects move without considering the forces that push them in that direction. It focuses on explaining how objects move through space and time, including their position, speed, and acceleration. Kinematics provides a framework for understanding and analyzing various types of motion, ranging from simple one-dimensional movements to more complex multidimensional and rotational motions.

Key Concepts In Kinematics

1. Position: This refers to the location of an object in space at a specific point in time. It is usually described in terms of coordinates, such as distance, displacement, or angle.
2. Displacement: The change in an object’s location from its initial point to its final point is represented by the vector quantity known as displacement. It takes into account both the direction and magnitude of the change.
3. Velocity: Velocity is a vector quantity that describes how fast an object’s position changes with respect to time.
4. Acceleration: An object’s rate of change in velocity with respect to time is measured by a vector quantity called acceleration. It indicates how quickly an object is speeding up or slowing down and in which direction.
5. Time: Time plays a crucial role in kinematics as it allows us to track how motion evolves over a specific period.

Also Check – Properties of Matter Formula

Kinematics Formula

Without taking into account the reason why it occurs, the Kinematics Formula is entirely concerned with the motion of bodies at specific places. To be exact, there are three kinematic formulas:

v=v_o+at

v²=v²₀+2a(x-x₀)

At this juncture,

x and x0 are Final and Initial displacements articulated in m,

v0 and v are initial and final velocity articulated in m/s,

acceleration is a and articulated in m/s2,

the time taken is t in s.

Also Check – Thermodynamics Formula

Kinematics Formulas for Projectile Motion

The kinematics formulas are:

In x-direction:

vx = v

x = x0 + vx0

In y-direction:

vy = vy0 – gt

y = y0 + vy₀t –1/2 gt²

Vy₂ = vy₀₂ – 2g(y – y₀)

These equations link five kinematic variables:

• Displacement (denoted by Δx)
• Initial Velocity v0
• Final Velocity denoted by v
• Time interval (denoted by t)
• Constant acceleration (denoted by a)

Also Check – Rotation Motion Formula

Kinematics Examples

Here are some examples of kinematics problems involving different types of motion:

1. Constant Velocity Motion: A car is traveling on a straight road at a constant speed of 60 km/h. How far will it travel in 2 hours?
2. Free Fall Motion: An object is dropped from a height of 20 meters. How long will it take for the object to reach the ground? (Assume no air resistance)
3. Projectile Motion: With a starting velocity of 15 m/s and an angle of 30 degrees above the horizontal, a ball is thrown off the edge of a cliff. How far will the ball land from the cliff’s edge?
4. Uniform Acceleration: A train is initially at rest and accelerates at a rate of 2 m/s². How long will it take for the train to reach a speed of 25 m/s?
5. Vertical Motion with Initial Velocity: A rocket is launched vertically upward at a 40 m/s initial speed. How far will the rocket travel before it begins to descend again?
6. Circular Motion: A car is moving around a circular track with a radius of 100 meters. If its speed is 20 m/s, How quickly is the automobile moving towards the circle’s centre?
7. Simple Harmonic Motion: A pendulum of length 1 meter is released from an angle of 30 degrees to the vertical. What is the pendulum’s bob’s top speed as it swings back and forth?
8. Relative Motion: Two cars are traveling on a straight road in the same direction. Car A is moving at 80 km/h, and Car B is moving at 100 km/h. If Car A is 200 meters ahead of Car B, how long will it take for Car B to catch up?

Inverse Kinematics:

Robotics and computer graphics employ inverse kinematics (IK) to calculate the joint angles or parameters necessary to position a robotic arm or a character’s body parts in a specific desired configuration. In simpler terms, it deals with the problem of figuring out how to position the individual components of a multi-jointed system to achieve a desired end effect, such as reaching a certain point in space or orienting an end effector in a particular direction.

In contrast to forward kinematics, which involves calculating the position and orientation of an end effector based on given joint angles, inverse kinematics focuses on calculating the joint angles required to achieve a specific end effector position and orientation. This is a crucial concept in robotics, animation, simulation, and various other fields where precise control over multi-jointed systems is needed.

Also Check – Work, Energy & Power Formula

There are four basic kinematics equations:

• v = v₀ + a t.
• Δ x = ( v + v ₀ ) t.
• Δ x = v ₀ t + 1 /2 a t ².
• v²= v²₀ + 2 a Δ x.

Rotational Kinematics Equations

Rotational kinematics deals with the motion of objects that are rotating around a fixed axis. Just like linear kinematics, which describes the motion of objects in a straight line, rotational kinematics provides equations to describe the relationships between angular displacement, angular velocity, angular acceleration, and time.

Here are the key rotational kinematics equations:

• Angular Displacement (θ):

• • • θ = ω_it + (1/2) αt2
    • where θ is the angular displacement, ω_i is the initial angular velocity, α is the angular acceleration, and t is time.

• Final Angular Velocity (ω_f):

• • • ω_f = ω_i + αt
    • where ω_f is the final angular velocity, ω_i is the initial angular velocity, α is the angular acceleration, and t is time.

• Angular Velocity (ω) and Time (t) Relationship:

• • • θ = ωt
    • where θ is the angular displacement, ω is the angular velocity, and t is time.

• Angular Acceleration (α) and Time (t) Relationship:

• • ω_f = ω_i + αt
  • where ω_f is the final angular velocity, ω_i is the initial angular velocity, α is the angular acceleration, and t is time.