1. Fundamental Concepts of Motion
Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move.
Position ($x$)
To describe motion, we first need to define an object's position. In one dimension, we establish a coordinate axis (e.g., the x-axis) with a designated origin ($x=0$) and a positive direction. The position of an object is its location on this axis, specified by a coordinate $x$. It is a vector quantity, where the sign indicates direction.
Displacement ($\Delta x$)
Displacement is the change in an object's position. If an object moves from an initial position $x_i$ at time $t_i$ to a final position $x_f$ at time $t_f$, the displacement is:
$$ \Delta x = x_f - x_i $$Displacement is a vector quantity, distinct from distance, which is the total scalar path length traveled.
Average Velocity ($\bar{v}$) and Average Speed
Average velocity is the ratio of displacement to the corresponding time interval. It is a vector quantity.
$$ \bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i} $$Average speed is a scalar quantity defined as the total distance traveled divided by the time interval.
$$ \text{Average Speed} = \frac{\text{Total Distance}}{\Delta t} $$2. Instantaneous Velocity and Speed
To describe velocity at a specific moment, we take the limit of the average velocity as the time interval $\Delta t$ approaches zero. This is the definition of the derivative.
Instantaneous velocity is the first time derivative of position:
$$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} $$Graphically, this is the slope of the tangent line to the position-time ($x-t$) graph. Instantaneous speed is the magnitude of the instantaneous velocity, $s = |v(t)|$.
3. Acceleration
Acceleration is the rate of change of velocity. It is the first time derivative of velocity and the second time derivative of position:
$$ a(t) = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2} $$Constant Acceleration and Kinematic Equations
For the special case of constant acceleration ($a$), we can derive the kinematic equations through integration. Here are the step-by-step derivations:
1. Velocity from Acceleration (Integration)
We start with the definition of constant acceleration, $a = \frac{dv}{dt}$. We rearrange and integrate, assuming an initial velocity $v_0$ at $t=0$ and velocity $v$ at time $t$:
$$ dv = a \, dt \implies \int_{v_0}^{v} dv' = \int_{0}^{t} a \, dt' $$Since $a$ is constant, it can be taken out of the integral:
$$ [v']_{v_0}^{v} = a \int_{0}^{t} dt' \implies v - v_0 = a[t']_{0}^{t} $$This yields the first kinematic equation:
2. Position from Velocity (Integration)
Next, we use $v = \frac{dx}{dt}$ and substitute our expression for $v$:
$$ \frac{dx}{dt} = v_0 + at $$Assuming an initial position $x_0$ at $t=0$ and position $x$ at time $t$, we separate variables and integrate:
$$ dx = (v_0 + at) \, dt \implies \int_{x_0}^{x} dx' = \int_{0}^{t} (v_0 + at') \, dt' $$ $$ [x']_{x_0}^{x} = \left[v_0t' + \frac{1}{2}at'^2\right]_0^t \implies x - x_0 = (v_0t + \frac{1}{2}at^2) - (0) $$This yields the second kinematic equation:
3. Time-Independent Equation (Algebraic Derivation)
We derive a relationship between velocity and displacement by eliminating time. From the first equation, $t = (v - v_0)/a$. We substitute this into the position equation:
$$ x - x_0 = v_0\left(\frac{v - v_0}{a}\right) + \frac{1}{2}a\left(\frac{v - v_0}{a}\right)^2 $$ $$ x - x_0 = \frac{v_0v - v_0^2}{a} + \frac{a}{2} \frac{v^2 - 2v_0v + v_0^2}{a^2} = \frac{v_0v - v_0^2}{a} + \frac{v^2 - 2v_0v + v_0^2}{2a} $$Multiplying by $2a$ to clear the denominators:
$$ 2a(x - x_0) = 2(v_0v - v_0^2) + (v^2 - 2v_0v + v_0^2) $$ $$ 2a(x - x_0) = 2v_0v - 2v_0^2 + v^2 - 2v_0v + v_0^2 $$Simplifying the expression gives the third kinematic equation:
General Motion via Integration
For a general, non-constant acceleration $a(t)$, we find velocity and position by direct integration:
$$ \Delta v = v_f - v_i = \int_{t_i}^{t_f} a(t) \, dt $$ $$ \Delta x = x_f - x_i = \int_{t_i}^{t_f} v(t) \, dt $$4. Problem Set & Solutions
Attempt to solve each problem in the text area provided, then click the button to reveal the detailed solution.