derive-s-ut-12at2-graphically
🚀 Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. One of the key concepts in kinematics is uniformly accelerated motion, where an object accelerates at a constant rate. The equation s = ut + (1/2)at² describes the displacement (s) of an object in uniformly accelerated motion, where u is the initial velocity, a is the acceleration, and t is the time. This equation can be derived graphically by analyzing the area under the velocity-time graph of the motion.
Theory Explanation
Understanding the Velocity-Time Graph
In a velocity-time graph, the velocity of an object is plotted on the y-axis and time on the x-axis. For uniformly accelerated motion, the graph is a straight line that slopes upwards if the object is accelerating. The area under the graph represents the displacement of the object during that time period.
Calculating the Area Under the Graph
For uniformly accelerated motion, the area under the velocity-time graph can be calculated as a trapezoid (or a triangle, depending on the initial velocity). The area A can be calculated using the formula A = (1/2) * (base) * (height). In this case, the base is time (t) and the height is the average velocity, which can be expressed as (u + v)/2, where v is the final velocity. Since v = u + at, we can express the average velocity as (u + (u + at))/2 = (2u + at)/2.
Relating Area to Displacement
The area calculated represents the displacement (s) of the object. Therefore, we can equate the area to displacement: s = (1/4) * t * (2u + at). By rearranging this equation and simplifying, we can derive the equation s = ut + (1/2)at².
Key Points
- 🎯 Uniformly accelerated motion involves constant acceleration.
- 🎯 The velocity-time graph for uniformly accelerated motion is linear.
- 🎯 The area under the velocity-time graph gives the displacement of the object.
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Examples:💡
An object starts from rest (u = 0) and accelerates at 2 m/s² for 5 seconds. Calculate the displacement.
Solution:
Step 1: Given: u = 0, a = 2 m/s², t = 5 s. Use the formula s = ut + (1/2)at².
Step 2: Calculate the displacement: s = 0 + (1/2) * 2 * 25 = 25 m.
Common Mistakes
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Mistake: Confusing the terms displacement and distance.
Correction: Displacement is a vector quantity that considers direction, while distance is a scalar quantity.
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Mistake: Using the wrong formula for displacement in uniformly accelerated motion.
Correction: Always remember to use s = ut + (1/2)at² for uniformly accelerated motion.