Derive position, velocity, and acceleration kinematics equations for an object undergoing a changing
acceleration?
I understand that somehow we need to get (∆a/∆t) as the change in acceleration into the kinematic equations but I really don't know where to start.
To derive the position, velocity, and acceleration kinematics equations for an object undergoing a changing acceleration, we need to start with the basic equations of motion. These equations relate the position, velocity, acceleration, and time for an object's motion.
Let's start with the basic equations of motion for an object moving in one dimension:
1. Position equation:
s = ut + (1/2)at^2
where:
s = position at time t
u = initial velocity
a = constant acceleration
t = time
2. Velocity equation:
v = u + at
where:
v = velocity at time t
3. Acceleration equation:
a = ∆v/∆t
where:
∆v = change in velocity
∆t = change in time
Now, if the object undergoes a changing acceleration (∆a/∆t), we can modify the acceleration equation to account for this change.
∆a/∆t represents the rate of change of acceleration with respect to time. It can be thought of as the derivative of acceleration with respect to time.
To include the changing acceleration in the equations, we can replace the constant acceleration 'a' in the original equations with an expression involving the changing acceleration (∆a/∆t):
1. Position equation:
s = ut + (1/2)(∆a/∆t)t^2
2. Velocity equation:
v = u + (∆a/∆t)t
3. Acceleration equation:
a = (∆a/∆t)
These derived equations now account for the changing acceleration of the object. Note that in order to calculate the precise values of position, velocity, and acceleration at any given time, you will need to know the initial conditions (initial position, initial velocity) and have information about how the acceleration is changing over time.