Kinematics: 1D & 2D Motion, Graphing, and Relative Velocity

Kinematics: 1D and 2D Motion

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. This section covers the analysis of motion in one and two dimensions, including the use of graphs, equations, and the concept of relative velocity.

1D Kinematic Equations

• Constant Acceleration: When an object moves with constant acceleration, its velocity and position can be described by the following equations:

$v = v_0 + at$

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

• Constant Velocity (a = 0): If there is no acceleration, the equations simplify to:

$v = v_0$

$x = x_0 + v_0 t$

• Key Terms:

  • x: Position

  • v: Velocity

  • a: Acceleration

  • t: Time

  • Subscripts 0: Initial values

Solving Kinematics Problems

• Many kinematics problems can be separated into parts due to:

  • Multiple moving objects

  • Different time periods with different behaviors

  • Multiple spatial coordinates (e.g., x and y)

• Strategy: Solve each part separately and connect them to find the final answer.

Example: A cheetah and a human start running at the same time. The cheetah catches up after a certain distance. Analyze each runner's motion separately and set their positions equal to find the time or distance at which they meet.

Graphical Analysis: x-t and v-t Graphs

• Position-Time (x-t) Graphs: Show how position changes over time.

• Velocity-Time (v-t) Graphs: Show how velocity changes over time.

• Key Concepts:

  • The slope of an x-t graph gives velocity.

  • The slope of a v-t graph gives acceleration.

  • The area under a v-t graph gives displacement.

Example: Given a position-time graph, determine which velocity-time graph corresponds to it by analyzing the slope (rate of change of position).

Calculus and Useful Pictures

• For more complex motion, such as changing velocity, use the area under the v-t graph to find displacement.

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

• The area under the v-t graph can be split into geometric shapes (e.g., rectangles and triangles) to calculate total displacement.

Example: If you accelerate from 3 m/s to 8 m/s over 10 seconds, the area under the v-t graph (a rectangle plus a triangle) gives the total distance traveled.

1D Kinematics Example: Kinesin Motor Protein

• A kinesin motor protein starts at rest, accelerates for 4.0 s, then moves at constant velocity for 6.0 s, covering a total distance of 100 μm.

• To find the acceleration during the first 4.0 s:

$\Delta x = A_1 + A_2$

$A_1 = \frac{1}{2} b h = \frac{1}{2} (4\,\text{s}) v_f$

$A_2 = (6\,\text{s}) v_f$

$\Delta x = (8\,\text{s}) v_f$

$v_f = a \times 4\,\text{s}$

$100\,\mu\text{m} = 32 a$

$a = 3.1\,\mu\text{m}/\text{s}^2$

• This problem can be solved using kinematic equations or by extracting information from a graph of the motion.

2D (Projectile) Motion

• Projectile motion involves two-dimensional motion under constant acceleration (usually gravity).

• The horizontal and vertical motions are analyzed separately:

$x: \quad a_x = 0, \quad v_{0x} = v_0 \cos \theta$

$y: \quad a_y = -g, \quad v_{0y} = v_0 \sin \theta$

• Key Equations:

$\Delta x = v_{x} \Delta t$

$\Delta y = v_{0y} \Delta t + \frac{1}{2} a_y (\Delta t)^2$

• To resolve the initial velocity into components:

$v_x = v \cos \theta$

$v_y = v \sin \theta$

• Example: You throw a hamster to a friend 10.0 m away at 11.0 m/s, 39° above horizontal, from a height of 2.0 m. Calculate the time of flight and the height when it reaches your friend.

Step 1: $\Delta x = v_x \Delta t$ Step 2: $\Delta y = v_{0y} \Delta t + \frac{1}{2} a_y (\Delta t)^2$ Step 3: $h = y_0 + \Delta y$

Result: The hamster reaches a height of 3.4 m above the ground when caught.

Concept Questions: Projectile Motion

• When a package is dropped from a plane moving at constant speed and altitude (ignoring air resistance), the package remains vertically under the plane while falling.

Relative Velocity

Introduction to Relative Velocity

Relative velocity describes the velocity of an object as observed from a particular reference frame. When two motions combine (e.g., a swimmer in a river), the total velocity is the vector sum of the individual velocities.

Key Concepts and Equations

• These problems typically involve three velocities (vectors):

  • Velocity of object relative to medium (e.g., swimmer relative to water)

  • Velocity of medium relative to ground (e.g., water relative to land)

  • Velocity of object relative to ground (e.g., swimmer relative to land)

• The vector equation is:

$\vec{v}_{SW} + \vec{v}_{WL} = \vec{v}_{SL}$

• Tip-to-tail vector addition is used to solve for the unknown velocity.

Examples of Relative Velocity

• Swimmer in a River: A swimmer swims across a river with a current. The resultant velocity is found by vector addition of the swimmer's velocity in still water and the velocity of the current.

• Ferryboat Crossing: A ferryboat crosses a channel with a current. The time to cross and the resultant velocity are found using the Pythagorean theorem and vector addition.

$|\vec{v}_{FG}| = \sqrt{v_{FW}^2 + v_{WG}^2}$

$t = \frac{d}{|\vec{v}_{FG}|}$

• Example: If a ferry can travel at 6 km/h in still water and the current is 4 km/h, the resultant speed is 4.47 km/h, and the time to cross a 1 km channel is 0.22 h (13 minutes).

Comments and Strategies for Relative Velocity Problems

• Identify which velocity is the 'wind speed' or 'water speed' (usually not the resultant).

• Combine the object's speed in still conditions with the medium's speed to find the resultant motion.

• Draw diagrams to visualize vector addition and solve for unknowns.

Practice Problems

• Launch a paper airplane towards a target with a crosswind. Determine the resultant velocity and the distance to the target.

• Try a more challenging problem where the wind is not perpendicular to the plane's motion.

Additional info: The kinesin motor protein example connects biological motion to kinematics, illustrating the universality of kinematic principles in both physics and biology.