The displacement \( x \) is proportional to \( t^2 \), i.e., \[ x \propto t^2 \]
Step-by-Step Solution:
1. Velocity Calculation:
Velocity \( v \) is the rate of change of displacement with respect to time: \[ v = \frac{dx}{dt} \] Since \( x \propto t^2 \), differentiating with respect to \( t \): \[ v \propto \frac{d}{dt}(t^2) \] \[ v \propto t \]
2. Conclusion:
The velocity \( v \) is directly proportional to \( t \).
Kinematics is a branch of physics that deals with the study of motion, without considering the forces that cause the motion. It is concerned with the position, velocity, and acceleration of objects, as well as the time it takes for them to move from one point to another.
In kinematics, the motion of an object is represented using mathematical equations and graphs. The most common quantities used in kinematics are displacement, velocity, acceleration, and time.
Displacement refers to the change in position of an object, and it is represented by the symbol Δx. Velocity is the rate at which an object's position changes over time, and it is represented by the symbol v. Acceleration is the rate at which an object's velocity changes over time, and it is represented by the symbol a.
By analyzing the relationships between these quantities, kinematics can be used to predict the motion of objects in different situations. For example, kinematics can be used to calculate the maximum height reached by a projectile, the distance traveled by a car during a given time, or the time it takes for a rollercoaster to complete a loop.
Kinematics is used in many areas of physics, including mechanics, astrophysics, and robotics. It is also used in engineering, where it is used to design and optimize the motion of machines and vehicles.
