Acceleration

Answer (a) moving with a uniform acceleration

The graph between displacement displacement and time for a particle moving uniform acceleration is a parabola.

Step 1: The slope of the velocity-time graph represents the acceleration of the body. In this case, the graph is a straight line making an angle of with the time axis. The slope of the line is given by: where .

Step 2: The value of is known to be . Therefore, the acceleration of the body is:
Step 3: Hence, the body is moving with an acceleration of m/s . Therefore, the correct answer is option (D).

We are given that the acceleration of the particle varies with time as: ### Step 1: Find the Velocity The relationship between acceleration and velocity is given by: Thus, we can integrate the acceleration to find the velocity: Here, is the constant of integration. Assuming the initial velocity at is zero: At : ### Step 2: Find the Displacement The relationship between velocity and displacement is given by: Thus, we can integrate the velocity to find the displacement: Here, is the constant of integration. Assuming the initial displacement at is zero: At : Thus, the displacement is and the velocity is . The correct answer is:

Step 1: Understanding the Concept:
Acceleration is defined as the rate of change of velocity with respect to time.
In graphical analysis, the rate of change of one variable with respect to another is represented by the slope of the tangent to the curve at that point.
Step 2: Key Formula or Approach:
The mathematical definition of instantaneous acceleration is: This derivative corresponds to the gradient or slope of a velocity ( ) versus time ( ) graph.
Step 3: Detailed Explanation:
In a velocity-time graph, the y-axis represents velocity and the x-axis represents time.
The slope ( ) of this graph is .
According to kinematics, this quantity is equal to the acceleration of the body.
For example, if the graph is a straight line, the acceleration is constant, and its value is given by the slope of that line.
Step 4: Final Answer:
The acceleration of a moving body is obtained from the slope of the velocity-time graph.