Motion Along a Line: Kinematics and Calculus in Physics

Motion Along a Line

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In this chapter, we focus on motion along a straight line, introducing key concepts such as position, velocity, and acceleration, and applying calculus to analyze motion.

Uniform Motion

Definition and Characteristics

• Uniform motion occurs when an object moves along a straight line at a constant speed.

• The position graph of uniform motion is a straight line; the slope of this line represents the object's velocity.

• Velocity is the rate of change of position with respect to time.

Equation of Uniform Motion:

$x_f = x_i + v_x (t_f - t_i)$

• $x_f$ = final position

• $x_i$ = initial position

• $v_x$ = constant velocity

• $(t_f - t_i)$ = time interval

Example: If a car moves at 5 m/s for 10 seconds, starting from $x_i = 0$, its final position is $x_f = 0 + 5 \times 10 = 50$ m.

Interpreting Position and Velocity Graphs

• Steeper slopes on a position-time graph indicate faster speeds.

• Negative slopes correspond to negative velocities (motion in the opposite direction).

• The velocity graph is constant for uniform motion.

Instantaneous Velocity and Calculus

Instantaneous Velocity

• Instantaneous velocity is the velocity of an object at a specific instant in time.

• It is given by the slope of the tangent to the position-time curve at that point.

Mathematical Definition:

$v_x = \frac{dx}{dt}$

• $\frac{dx}{dt}$ is the derivative of position with respect to time.

Example: If $x(t) = 3t^2 - 2t + 1$, then $v_x = \frac{d}{dt}(3t^2 - 2t + 1) = 6t - 2$.

Derivatives and Their Rules

• The derivative of a constant is zero.

• The derivative of a sum is the sum of the derivatives.

• For $x = ct^n$, $\frac{dx}{dt} = nct^{n-1}$.

Finding Position from Velocity: Integrals

Displacement as Area Under the Curve

• The displacement of an object is the area under the velocity-time graph.

• For variable velocity, use integration:

$\Delta x = \int_{t_i}^{t_f} v_x \, dt$

• For constant velocity: $\Delta x = v_x (t_f - t_i)$

Example: If $v_x = 2t$, then $\Delta x = \int_0^3 2t \, dt = [t^2]_0^3 = 9$ m.

Constant Acceleration

Definition and Equations

• Acceleration is the rate of change of velocity with respect to time.

• For constant acceleration $a_x$:

$a_x = \frac{dv_x}{dt}$

$v_x = v_{xi} + a_x (t - t_i)$

$x = x_i + v_{xi}(t - t_i) + \frac{1}{2} a_x (t - t_i)^2$

$v_x^2 = v_{xi}^2 + 2a_x (x - x_i)$

• $v_{xi}$ = initial velocity

• $a_x$ = constant acceleration

Example: A car accelerates from rest ($v_{xi} = 0$) at $a_x = 2$ m/s2 for 5 s. Final velocity: $v_x = 0 + 2 \times 5 = 10$ m/s.

Sign of Acceleration

• Positive acceleration: speeding up in the positive direction or slowing down in the negative direction.

• Negative acceleration: slowing down in the positive direction or speeding up in the negative direction.

Solving Kinematic Problems

Problem-Solving Strategy

1. Prepare: Model the situation, draw diagrams, define variables.

2. Solve: Use appropriate kinematic equations.

3. Assess: Check units, significant figures, and reasonableness of the answer.

Example: A rocket launches with $a_x = 18$ m/s2 for 150 s. Final velocity: $v_x = 0 + 18 \times 150 = 2700$ m/s. Distance: $x = 0 + 0 \times 150 + 0.5 \times 18 \times (150)^2 = 202,500$ m.

Free Fall

Definition and Properties

• Free fall is motion under the influence of gravity alone.

• All objects in free fall have the same acceleration, $g$.

• Standard value: $g = 9.8$ m/s2 (downward).

• Use kinematic equations with $a_y = -g$ for vertical motion.

Example: Dropping a rock from 100 m: $y = y_0 + v_{y0} t - \frac{1}{2} g t^2$. With $v_{y0} = 0$, solve for $t$ when $y = 0$.

Projectile Motion

Definition and Components

• A projectile moves in two dimensions under gravity.

• Initial velocity can be split into horizontal ($v_{x0}$) and vertical ($v_{y0}$) components:

$v_{x0} = v_0 \cos \theta$

$v_{y0} = v_0 \sin \theta$

• Horizontal motion: constant velocity ($a_x = 0$)

• Vertical motion: constant acceleration ($a_y = -g$)

Equations:

Horizontal: $x = x_0 + v_{x0} t$

Vertical: $y = y_0 + v_{y0} t - \frac{1}{2} g t^2$

Range (for launch and landing at same height):

$R = \frac{v_0^2 \sin 2\theta}{g}$

• Maximum range at $\theta = 45^\circ$.

Example: A ball is launched at $v_0 = 10$ m/s, $\theta = 30^\circ$. $v_{x0} = 10 \cos 30^\circ = 8.66$ m/s, $v_{y0} = 10 \sin 30^\circ = 5$ m/s.

Modeling Changing Acceleration

Instantaneous Acceleration and Graphs

• Instantaneous acceleration is the slope of the velocity-time graph at a point.

• Velocity can be found by integrating the acceleration graph.

Exponential Functions in Motion

• Some motions (e.g., sprinters, animals) are modeled with exponential functions:

$u = A e^{-t/\tau}$

• $\tau$ is the time constant; after time $\tau$, the quantity decays to 37% of its initial value.

Example: A horse accelerates with $a_{max} = 5.7$ m/s2, $v_{max} = 86$ km/h. The time constant $\tau$ relates $a_{max}$ and $v_{max}$ by $v_{max} = a_{max} \tau$.

Summary Table: Kinematic Quantities and Equations

Quantity	Definition	Equation
Position ($x$)	Location along a line	--
Displacement ($\Delta x$)	Change in position	$\Delta x = x_f - x_i$
Velocity ($v_x$)	Rate of change of position	$v_x = \frac{dx}{dt}$
Acceleration ($a_x$)	Rate of change of velocity	$a_x = \frac{dv_x}{dt}$
Uniform motion	Constant velocity	$x_f = x_i + v_x (t_f - t_i)$
Constant acceleration	Constant rate of change of velocity	$v_x = v_{xi} + a_x (t - t_i)$ $x = x_i + v_{xi}(t - t_i) + \frac{1}{2} a_x (t - t_i)^2$ $v_x^2 = v_{xi}^2 + 2a_x (x - x_i)$
Free fall	Motion under gravity	$a_y = -g$
Projectile range	Horizontal distance traveled	$R = \frac{v_0^2 \sin 2\theta}{g}$

Key Takeaways

• Kinematics uses position, velocity, and acceleration to describe motion.

• Calculus (derivatives and integrals) provides precise definitions and tools for analyzing motion.

• Constant acceleration allows for the use of kinematic equations to solve a wide range of problems, including free fall and projectile motion.

• Graphical analysis (slopes and areas) is essential for interpreting and solving motion problems.