derive second equation of motion.S=ut+1/2at2 graphycally where the symbols have their usual meanings.
To derive the second equation of motion, we need to relate the displacement (S) of an object to its initial velocity (u), acceleration (a), and time (t). The equation you mentioned, S = ut + 1/2at^2, is one of the equations of motion in physics.
To understand how this equation is derived graphically, we can start by plotting a graph that represents the motion of the object. Let's assume that the object starts from rest, meaning its initial velocity, u, is 0. This simplifies our graph to a distance-time graph.
On the y-axis, we plot the displacement, S, and on the x-axis, we plot the time, t. Now, let's analyze the different factors in the equation and how they contribute to the graph.
1. Initial Position (S₀): The graph starts at the origin (S₀ = 0) since the object starts from rest.
2. Velocity-Time Graph: The first term in the equation, ut, represents the area under a line graph of velocity plotted against time. In our case, since the initial velocity is 0, this term does not contribute to the graph.
3. Acceleration-Time Graph: The second term in the equation, 1/2at^2, represents the area under a line graph of acceleration plotted against time squared. This term contributes to the shape of the graph as it represents the changing acceleration over time.
The area under the acceleration-time graph up to time t is given by the shaded region in the graph.
graph TD
A((Origin))
B((Line graph))
C((Area under the graph))
A --> B
B --> C
By calculating this shaded area, we can determine the displacement, S, at time t.
To find the equation, we can use calculus to find the analytical expression for the area under the graph of acceleration.
For constant acceleration, a, the acceleration-time graph is a straight line. The area under a straight line is given by the formula: Area = 1/2 * base * height.
In this case, the base is t, and the height is a. Therefore, the area under the acceleration-time graph is 1/2 * at^2.
This is the contribution of the second term in the equation, 1/2at^2, to the displacement, S.
Finally, the complete equation, S = ut + 1/2at^2, is derived graphically by considering the area under the acceleration-time graph and accounting for the initial position.
Please note that this explanation is a simplified graphical derivation. The more rigorous derivation involves the use of calculus and kinematic equations.