Kinematics: Motion at Constant Acceleration and Free Fall

Chapter 2: Motion at Constant Acceleration

Acceleration and Deceleration

Understanding acceleration is crucial in kinematics, especially when distinguishing between speeding up and slowing down (deceleration). The direction of the velocity and acceleration vectors determines whether an object speeds up or slows down.

• Speeding Up: If the velocity vector (v) and the acceleration vector (a) point in the same direction (i.e., have the same sign in one dimension), the object speeds up.

• Slowing Down (Decelerating): If v and a point in opposite directions (i.e., have opposite signs in one dimension), the object slows down.

Example: If a car is moving to the right (+x direction) and its acceleration is also to the right, it speeds up. If the acceleration is to the left while the car moves right, it slows down.

Motion at Constant Acceleration

When an object moves with constant acceleration, its position and velocity change predictably over time. In one dimension, vector notation is often omitted, but direction is indicated by sign.

• Displacement: $\Delta x = x_2 - x_1$

• Average Velocity: $\bar{v} = \frac{x_2 - x_1}{t_2 - t_1} = \frac{\Delta x}{\Delta t}$

• Average Acceleration: $\bar{a} = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t}$

For constant acceleration, the average acceleration equals the instantaneous acceleration at all times.

• $\bar{a} = a$

• Average velocity for constant acceleration: $\bar{v} = \frac{v_0 + v}{2}$

Constant Acceleration Kinematic Equations (One Dimension)

These equations describe the position and velocity of an object as functions of time under constant acceleration:

• $v = v_0 + at$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a(x - x_0)$

• $\bar{v} = \frac{v_0 + v}{2}$

Alternative forms:

• $\Delta v = a t$

• $\Delta x = v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a \Delta x$

Note: If the clock starts at $t_0 \neq 0$, replace $t$ by $\Delta t$ in these equations. For vertical motion, use $y$ instead of $x$.

General Strategy for Solving Kinematics Problems

Solving kinematics problems systematically improves accuracy and understanding.

1. Draw a diagram of the physical situation, labeling known and unknown quantities and indicating positive directions.

2. List all known quantities (symbols and values) and unknowns (what you need to find).

3. Identify relevant physical concepts and write down the corresponding general equations.

4. Use a single equation if possible. If information is missing, use more than one equation.

5. Work symbolically, substituting numbers only at the end to check for errors and unit consistency.

6. Check your answer for reasonableness and correct units.

If motion is divided into segments, the final position and velocity of one segment become the initial conditions for the next. Sometimes, two solutions may arise; only physically reasonable ones should be accepted (e.g., positive time).

Worked Example: Boat with Constant Acceleration

Problem: A boat has an initial velocity of $+6.0\ \text{m/s}$ and an acceleration of $+2.0\ \text{m/s}^2$. What is its velocity after $8.0\ \text{s}$? How far did it travel?

• Known: $a = 2.0\ \text{m/s}^2$, $v_0 = 6.0\ \text{m/s}$, $t = 8.0\ \text{s}$, $x_0 = 0$

• Unknown: $v$, $x$

Solution:

• $v = v_0 + at = 6.0 + (2.0)(8.0) = 22\ \text{m/s}$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2 = 0 + (6.0)(8.0) + \frac{1}{2}(2.0)(8.0)^2 = 48 + 64 = 112\ \text{m}$

Worked Example: Jet on a Runway

Problem: A jet starts from rest and travels down a runway with acceleration $+3.1\ \text{m/s}^2$ and reaches a velocity of $+62\ \text{m/s}$. It then travels at this velocity for $6.0\ \text{s}$ before takeoff. What is the total distance travelled?

• Stage 1: Accelerating from rest ($v_0 = 0$) to $v_1 = 62\ \text{m/s}$ with $a_1 = 3.1\ \text{m/s}^2$

• Stage 2: Constant velocity $v_2 = 62\ \text{m/s}$ for $t_2 = 6.0\ \text{s}$

Solution:

• Stage 1: $x_1 = \frac{v_1^2}{2a_1} = \frac{(62)^2}{2 \times 3.1} = 620\ \text{m}$

• Stage 2: $x_2 = v_2 t_2 = (62)(6.0) = 372\ \text{m}$

• Total distance: $x = x_1 + x_2 = 620 + 372 = 992\ \text{m}$

• Additional info: The notes use $x_2 = x_1 + v_1 t_2$ for the second stage, which matches the above calculation.

Freely Falling Objects

Free Fall and Acceleration Due to Gravity

Galileo's experiments showed that, in the absence of air resistance, all objects fall with the same constant acceleration near Earth's surface. This idealized motion is called free fall.

• Acceleration due to gravity: $g = 9.80\ \text{m/s}^2$ (always points downward)

• In free fall, the kinematic equations apply with $a = -g$ (if upward is positive).

Example: A stone is dropped from a tall building. After $3.00\ \text{s}$ of free fall, what is its displacement?

• Known: $v_0 = 0$, $t = 3.0\ \text{s}$, $a = -9.80\ \text{m/s}^2$

• Unknown: $\Delta y$

• Solution: $\Delta y = v_0 t + \frac{1}{2} a t^2 = 0 + \frac{1}{2}(-9.80)(3.00)^2 = -44.1\ \text{m}$

Summary Table: Kinematic Equations for Constant Acceleration

Equation	Physical Meaning
$v = v_0 + at$	Velocity as a function of time
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	Position as a function of time
$v^2 = v_0^2 + 2a(x - x_0)$	Velocity as a function of position
$\bar{v} = \frac{v_0 + v}{2}$	Average velocity for constant acceleration
$g = 9.80\ \text{m/s}^2$	Acceleration due to gravity (free fall)

Key Terms and Concepts

• Acceleration: Rate of change of velocity; can be positive (speeding up) or negative (slowing down).

• Deceleration: Slowing down; occurs when acceleration is opposite to velocity.

• Displacement: Change in position; $\Delta x$ or $\Delta y$.

• Free Fall: Motion under gravity alone, with $a = -g$.

• Kinematic Equations: Set of equations describing motion under constant acceleration.

Applications

• Solving problems involving cars, boats, jets, and falling objects using kinematic equations.

• Analyzing motion in segments when acceleration changes or when velocity becomes constant.

• Checking for physically reasonable solutions (e.g., positive time, correct direction).

Additional info: These notes provide a foundation for more advanced topics in kinematics and dynamics, including projectile motion and multidimensional analysis.