Introduction

Kinematics is the branch of physics that studies motion. It helps us understand how objects move from one place to another. Whether it’s a car traveling down the road or a baseball being thrown, kinematics explains these motions. This review covers key kinematics formulas you’ll need in AP® Physics 1, along with practical examples to see them in action.

Understanding Motion

Motion refers to the change in position of an object over time. It’s a fundamental concept because it applies to everything—from planets orbiting the sun to the mechanics of a race car. Motion can be represented using diagrams, graphs, and equations. Imagine a car driving from one city to another. The path it takes, its speed, and how quickly it accelerates are all aspects of motion.

• Example: Consider a car that starts from rest and gradually speeds up. This scenario can be analyzed using kinematics formulas to determine its speed and position over time.

Key Concepts in Kinematics

Displacement vs. Distance

• Displacement is the shortest path between two points and includes direction. Displacement is a vector.
• Distance is the total path traveled, regardless of direction. Distance is a scalar.
• Example: Walking once around a park covering 500 meters means your distance is 500 meters, but if you start and return to the same spot, your displacement is zero.

Velocity vs. Speed

• Velocity is speed with a direction. Velocity, like displacement, is a vector.
• Speed is how fast an object is moving, without considering direction. Like distance, speed is a scalar.
• Average speed is related to the total distance divided by the time and average velocity is related to the average displacement divided by the time.
• Example: If a car is going 60 km/h East, then 60 km/h is its speed (without direction) and 60 km/h East is its velocity.

Acceleration

• Acceleration is the rate at which velocity changes.
• Average and instantaneous acceleration differ, with average being over time and instantaneous at a specific moment.
• Example: A skateboarder pushing off to speed up shows positive acceleration, while slowing to a stop shows negative acceleration.

The Kinematic Equations

These kinematic formulas apply to objects moving with constant acceleration:

1. v = v_0 + a \cdot t
2. x = x_0 + v_0 \cdot t + \frac{1}{2} a \cdot t^2
3. v^2 = v_0^2 + 2a(x - x_0)

Here, v is final velocity, v_0 is initial velocity, a is acceleration, x is final position, and x_0 is initial position.

Applying Kinematic Formulas

Example 1: A car starts from rest and accelerates at 3, \text{m/s}^2 for 5 seconds. Find the distance traveled.

Solution Steps:

• Given: v_0 = 0 \text{ m/s}, a = 3 \text{ m/s}^2, t = 5 \text{ s}.
• Use x = x_0 + v_0 \cdot t + \frac{1}{2} a \cdot t^2.
• Calculate: x = 0 + 0 \cdot 5 + \frac{1}{2} \cdot 3 \cdot (5)^2 = 37.5 \text{ m}.

Understanding Motion Graphs

Motion graphs are crucial for visualizing how objects move over time:

• Position-Time Graph: This shows how position changes over time.
• Velocity-Time Graph: Illustrates velocity changes.
• Acceleration-Time Graph: Highlights changes in acceleration.

Two of the most important graphs in physics are position-time (p-t) graphs and velocity-time (v-t) graphs. Each feature of these graphs—such as slope and the area under the curve—tells us something about an object’s motion.

Position-Time Graphs

A position-time graph plots an object’s location against time. Here’s what different features mean:

• Slope = Velocity: The steeper the slope, the faster the object moves. A horizontal line means zero velocity (the object is at rest).
• Curvature = Acceleration: A straight line means constant velocity. A curved graph indicates changing velocity, meaning acceleration is present.
• Common Shapes:
  • A straight diagonal line represents constant velocity.
  • A parabola (curved graph) means acceleration is occurring. A concave-up parabola means increasing velocity, while a concave-down one means decreasing velocity.

Since the slope of a position-time graph represents velocity, a parabolic p-t graph corresponds to a linear v-t graph.

Velocity-Time Graphs

A velocity-time graph shows how an object’s velocity changes over time.

• Slope = Acceleration: The slope of a v-t graph gives the acceleration. A steep slope means rapid acceleration, while a flat line means constant velocity (zero acceleration).
• Area Under the Curve = Displacement: The total area under a v-t graph gives the object’s displacement (change in position).
  • If the graph forms a rectangle: \text{Displacement}=\text{base}\times\text{height}
  • If it forms a triangle: \text{Displacement}=\frac{1}{2}\times \text{base}\times \text{height}
  • Sometimes, you might need to combine multiple shapes and formulas to calculate the full area.
• Common Shapes:
  • A horizontal line means constant velocity.
  • A straight diagonal line means constant acceleration.

Since the slope of a v-t graph represents acceleration, a linear v-t graph corresponds to a constant acceleration motion.

By analyzing these graphs, we can determine how an object moves, even without knowing its exact equations!

The Role of Gravity in Kinematics

Gravity affects motion by pulling objects towards Earth at a_g \approx 9.81 \text{ m/s}^2. When objects are free-falling, they accelerate at this rate. Often, this rate is approximated to 10\text{ m/s}^2.

Example 3: A ball is thrown upwards. How high does it go?

Given: Initial velocity v_0 = 20\text{ m/s}.

• Use v^2 = v_0^2 + 2a(x - x_0) to find maximum height.
• Final velocity at peak (v = 0).

• Solve: x = 20 \text{ m}.

Summary of Key Concepts

Conclusion

To excel in AP® Physics, practice applying these concepts and solving related problems. Mastering kinematics formulas is not just about memorizing formulas, but also understanding motion and predicting outcomes. Keep tackling new problems and explore resources like videos and practice tests to reinforce your knowledge.

Sharpen Your Skills for AP® Physics 1

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