If you have studied physics, you must have come across the equation v^2 = u^2 + 2as. This formula is used to calculate the final velocity (v) of an object in motion, given the initial velocity (u), acceleration (a), and displacement (s).
What is Velocity?
Velocity is the rate of change of an object's position with respect to time. It is a vector quantity that has both magnitude and direction. In simpler terms, it tells us how fast an object is moving and in which direction.
What is Acceleration?
Acceleration, on the other hand, is the rate of change of an object's velocity with respect to time. It is also a vector quantity that has both magnitude and direction. It tells us how much an object's velocity changes in a given amount of time.
What is Displacement?
Displacement is the change in position of an object during motion. It is a vector quantity that has both magnitude and direction. It tells us how far an object has moved from its starting point and in which direction.
Deriving the Formula
The formula v^2 = u^2 + 2as can be derived using the equations of motion. The first equation of motion is:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. Rearranging this equation, we get:
t = (v - u) / a
Substituting this value of t in the second equation of motion:
s = ut + 1/2at^2
we get:
s = u((v - u) / a) + 1/2a((v - u) / a)^2
Simplifying this equation, we get:
s = ((v^2 - u^2) / 2a)
Multiplying both sides by 2a, we get:
2as = v^2 - u^2
Adding u^2 to both sides, we get:
v^2 = u^2 + 2as
Interpreting the Formula
The formula v^2 = u^2 + 2as tells us that the final velocity of an object in motion depends on its initial velocity, acceleration, and displacement. If an object starts with an initial velocity u and undergoes an acceleration a for a displacement s, its final velocity v can be calculated using this formula.
Examples of Using the Formula
Let's take a few examples to understand how the formula v^2 = u^2 + 2as can be used to calculate the final velocity of an object in motion.
Example 1: An object is thrown upwards with an initial velocity of 20 m/s. It reaches a maximum height of 50 m. What is its final velocity?
Here, u = 20 m/s, s = 50 m, and a = -9.8 m/s^2 (the acceleration due to gravity). Substituting these values in the formula v^2 = u^2 + 2as, we get:
v^2 = (20)^2 + 2(-9.8)(50) = 400 - 980 = -580
Since velocity cannot be negative, we take the square root of both sides to get:
v = 24.08 m/s (approx)
Example 2: A car accelerates from rest at a rate of 5 m/s^2. It covers a distance of 100 m. What is its final velocity?
Here, u = 0 m/s, s = 100 m, and a = 5 m/s^2. Substituting these values in the formula v^2 = u^2 + 2as, we get:
v^2 = (0)^2 + 2(5)(100) = 1000
Taking the square root of both sides, we get:
v = 31.62 m/s (approx)
Conclusion
The formula v^2 = u^2 + 2as is an important equation in physics that helps us calculate the final velocity of an object in motion. It is derived using the equations of motion and depends on the initial velocity, acceleration, and displacement of the object. Understanding this formula can help us solve problems related to motion and velocity.
