Kinematics with constant acceleration is all about predicting motion using math. We use equations to figure out how things move when they speed up or slow down at a steady rate.

These equations help us calculate position, velocity, and acceleration over time. They're super useful for real-world problems, like figuring out how far a car will travel or how high a ball will go when thrown.

Kinematics with Constant Acceleration

Kinematic equations from calculus

• Acceleration represents the rate of change of velocity with respect to time and is expressed as $a = \frac{dv}{dt}$
• Velocity represents the rate of change of position with respect to time and is expressed as $v = \frac{dx}{dt}$
• Velocity as a function of time is derived by integrating acceleration with respect to time resulting in $v(t) = \int a dt = at + v_0$, where $v_0$ represents the initial velocity (at $t=0$)
• Position as a function of time is derived by integrating velocity with respect to time resulting in $x(t) = \int v(t) dt = \int (at + v_0) dt = \frac{1}{2}at^2 + v_0t + x_0$, where $x_0$ represents the initial position (at $t=0$)

Application of kinematic equations

• The equation $v(t) = v_0 + at$ is used to find velocity at a specific time by substituting known values for initial velocity $v_0$, acceleration $a$, and time $t$ (car accelerating from rest)
• The equation $x(t) = x_0 + v_0t + \frac{1}{2}at^2$ is used to find position at a specific time by substituting known values for initial position $x_0$, initial velocity $v_0$, acceleration $a$, and time $t$ (ball thrown upwards)
• The equation $v_f^2 = v_0^2 + 2a(x_f - x_0)$ relates final velocity $v_f$, initial velocity $v_0$, acceleration $a$, and displacement $x_f - x_0$ and is derived by substituting $t$ from $v(t) = v_0 + at$ into $x(t) = x_0 + v_0t + \frac{1}{2}at^2$ (braking distance of a car)
• These equations describe the motion of objects and their trajectory over time

Velocity functions from acceleration

• For constant acceleration, the equation $v(t) = v_0 + at$ is used by substituting the given constant acceleration value for $a$ and including the initial velocity $v_0$ if provided, otherwise assuming $v_0 = 0$ (object falling under gravity)
• For acceleration as a function of time, the velocity function is obtained by integrating the acceleration function with respect to time using $v(t) = \int a(t) dt + v_0$, evaluating the integral and adding the initial velocity $v_0$ if provided, otherwise assuming $v_0 = 0$ (rocket with varying thrust)
• Initial conditions, such as initial velocity and position, are crucial for determining the complete velocity function

Position functions from velocity

• For constant velocity, the equation $x(t) = x_0 + vt$ is used by substituting the given constant velocity value for $v$ and including the initial position $x_0$ if provided, otherwise assuming $x_0 = 0$ (train moving at constant speed)
• For velocity as a function of time, the position function is obtained by integrating the velocity function with respect to time using $x(t) = \int v(t) dt + x_0$, evaluating the integral and adding the initial position $x_0$ if provided, otherwise assuming $x_0 = 0$ (object thrown with varying velocity)

Vector and Scalar Quantities in Kinematics

• Velocity and acceleration are vector quantities, having both magnitude and direction
• Time and displacement are scalar quantities, having only magnitude
• Understanding the distinction between vector and scalar quantities is essential for correctly applying kinematic equations