Motion Along a Straight Line: Study Notes

Motion Along a Straight Line

Review of Vectors and Scalars

In physics, quantities are classified as either vectors or scalars based on whether they possess direction in addition to magnitude.

• Scalar: A quantity described only by its magnitude (size). Examples: temperature, distance, speed.

• Vector: A quantity described by both magnitude and direction. Examples: displacement, velocity, force.

Examples:

• Temperature: Scalar ("It's 60°F outside")

• Force: Vector ("I pushed with 100N north")

• Distance: Scalar ("I walked for 10 m")

• Displacement: Vector ("I walked 10 m east")

• Speed: Scalar ("I drove at 80 mph")

• Velocity: Vector ("I drove 80 mph west")

Distance, Displacement, Speed, and Velocity

Distance is the total length of the path traveled, while displacement is the straight-line change in position from the initial to the final point. Speed is the rate at which distance is covered, and velocity is the rate at which displacement changes, including direction.

• Distance (d): Always positive; scalar.

• Displacement (\( \Delta x \)): Can be positive or negative; vector.

• Speed: Always positive; scalar.

• Velocity: Can be positive or negative; vector.

Formulas:

• Average speed: $\text{speed} = \frac{\text{distance}}{\text{time}}$

• Average velocity: $\vec{v}_{\text{avg}} = \frac{\Delta x}{\Delta t}$

Example: If you jog 15 m in 2 s, then 9 m backwards in another 2 s, calculate your speed and velocity for the total trip.

Solving Constant and Average Velocity Problems

When velocity is constant, the average velocity equals the instantaneous velocity. The main equation used is:

• $\vec{v}_{\text{avg}} = \frac{\Delta x}{\Delta t}$

• Rearranged: $\Delta x = \vec{v}_{\text{avg}} \Delta t$

Example: If $x_0 = 0$ m, $x = 20$ m, $\Delta t = 5$ s, then $v = ?$

Constant Velocity with Multiple Parts

When an object moves with different constant velocities in different intervals, solve each part separately and combine results for total distance and average velocity.

• For each interval: $\Delta x = v \Delta t$

• Total distance = sum of all $\Delta x$

• Average velocity = $\frac{\text{total displacement}}{\text{total time}}$

Steps:

1. Draw a diagram and list variables for each interval.

2. Write equations for each interval.

3. Solve for unknowns.

Introduction to Acceleration

Acceleration is the rate at which velocity changes with time. It is always a vector quantity.

• Formula: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

• Units: $\text{m/s}^2$

• Acceleration can result from a change in the magnitude or direction of velocity.

Example: If your car moves right at 10 m/s and after 4 s is moving right at 30 m/s, the acceleration is $\vec{a} = \frac{30 - 10}{4} = 5$ m/s$^2$ to the right.

Position-Time Graphs and Velocity

Position-time graphs plot position ($x$) on the y-axis and time ($t$) on the x-axis. The slope of the graph at any point gives the velocity.

• Upward slope: moving forward (positive velocity)

• Flat slope: stopped (zero velocity)

• Downward slope: moving backward (negative velocity)

Average velocity: $\vec{v}_{\text{avg}} = \frac{\Delta x}{\Delta t}$ (slope between two points)

Instantaneous velocity: Slope of the tangent line at a single point.

Curved Position-Time Graphs and Acceleration

If the position-time graph is curved, the velocity is changing, indicating nonzero acceleration.

• Curving up (concave up): positive acceleration

• Curving down (concave down): negative acceleration

• Straight lines: constant velocity, zero acceleration

Velocity-Time Graphs and Acceleration

Velocity-time graphs plot velocity on the y-axis and time on the x-axis. The slope of the graph gives the acceleration.

• Average acceleration: $a_{\text{avg}} = \frac{\Delta v}{\Delta t}$

• Instantaneous acceleration: Slope of the tangent line at a point

• Area under the velocity-time graph gives displacement: $\Delta x = \text{area under } v-t \text{ curve}$

Calculating Displacement from Velocity-Time Graphs

The area under the velocity-time graph between two points in time represents the displacement during that interval.

• Area above the time axis: positive displacement

• Area below the time axis: negative displacement

• For rectangles: $A = b \times h$

• For triangles: $A = \frac{1}{2} b h$

Equations of Motion (Kinematics Equations)

For constant acceleration, use the following equations (Uniformly Accelerated Motion, UAM):

• (1) $v = v_0 + a t$

• (2) $v^2 = v_0^2 + 2 a \Delta x$

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

• (4) $\Delta x = \frac{v_0 + v}{2} t$

To solve problems, identify the known and unknown variables, select the appropriate equation, and solve for the target variable.

Vertical Motion and Free Fall

Objects in free fall experience constant acceleration due to gravity ($g = 9.8$ m/s$^2$ downward). The same kinematics equations apply, with $a_y = -g$ if up is positive.

• Vertical equations mirror horizontal ones, with $y$ replacing $x$ and $a_y$ replacing $a_x$.

• Example: Dropping a ball from rest, $v_y = v_{0y} + a_y t$

Solving "Catch Up" or "Overtake" Problems

When one object catches up to another, set their position equations equal at the same time and solve for the unknowns.

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

• Set $x_A = x_B$ and solve for $t$ or $x$ as needed.

Summary Table: Scalars vs. Vectors

Quantity	Magnitude?	Direction?	Vector/Scalar
Temperature	Yes	No	Scalar
Force	Yes	Yes	Vector
Distance	Yes	No	Scalar
Displacement	Yes	Yes	Vector
Speed	Yes	No	Scalar
Velocity	Yes	Yes	Vector

Additional info: This guide covers the foundational concepts of one-dimensional motion, including graphical analysis, equations of motion, and problem-solving strategies for both constant velocity and constant acceleration scenarios. Practice problems and step-by-step solutions are essential for mastering these topics.