3.6 Finding Velocity and Displacement from Acceleration

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