unit 1: Kinematics
Overview
Kinematics is the branch of mechanics that deals with the features and properties of objects in motion. Using the three kinematics equations, we can find the correlation between the variables Xi, Xf, Vi, Vf, Δt, a. Velocity is how fast the position is changing over time. Acceleration is how fast the velocity is changing over time. The position graph provides the foundation for the velocity-time graphs and acceleration-time graphs.
Kinematics is the branch of mechanics that deals with the features and properties of objects in motion. Using the three kinematics equations, we can find the correlation between the variables Xi, Xf, Vi, Vf, Δt, a. Velocity is how fast the position is changing over time. Acceleration is how fast the velocity is changing over time. The position graph provides the foundation for the velocity-time graphs and acceleration-time graphs.
Terminology
Position - the position x is where an object is at time t. It can be solved by subtracting the final position by the initial position.
Distance - distance is how far someone or something has traveled in a set amount of time without considering direction. The distance traveled will always be positive.
Displacement - displacement is the overall change in position that determines how far someone or something is from the initial position. Displacement can have a direction, and it can be both positive and negative.
Position - the position x is where an object is at time t. It can be solved by subtracting the final position by the initial position.
Distance - distance is how far someone or something has traveled in a set amount of time without considering direction. The distance traveled will always be positive.
Displacement - displacement is the overall change in position that determines how far someone or something is from the initial position. Displacement can have a direction, and it can be both positive and negative.
Interpreting Position Time Graph
A position time graph shows the position x of an object at time t. The x-axis is time in seconds, while the y-axis is position in meters. A point on the graph would be graphed as (t,x), where the x value is t and the y value is x. For a linear model, as time increases position is increasing at a constant rate. In this specific position graph, the position is increasing 5 meters every increase in 1 second. Different position time graphs have different curves or lines that define the relationship between time and position. If you're given the slope of a linear graph, you can determine the position simply by multiple the slope by the time. The steepness of the slope demonstrates how fast an object is moving, while the sign of slope shows the direction that the position is moving towards. The y-intercept of the position time graph tells us the initial position
A position time graph shows the position x of an object at time t. The x-axis is time in seconds, while the y-axis is position in meters. A point on the graph would be graphed as (t,x), where the x value is t and the y value is x. For a linear model, as time increases position is increasing at a constant rate. In this specific position graph, the position is increasing 5 meters every increase in 1 second. Different position time graphs have different curves or lines that define the relationship between time and position. If you're given the slope of a linear graph, you can determine the position simply by multiple the slope by the time. The steepness of the slope demonstrates how fast an object is moving, while the sign of slope shows the direction that the position is moving towards. The y-intercept of the position time graph tells us the initial position
Interpreting Velocity Time Graphs
Velocity is how fast an object is changing its position with respect to time. The x-axis is time and the y-axis is velocity in m/s. If velocity is positive, the object is moving away from the point of origin. If the velocity is negative, the object is moving towards the point of origin. We can also interpret velocity time graphs through the slope. When the slope is positive, the acceleration is positive, while when the slope is negative the acceleration is negative. Since acceleration is just the change in velocity over time, we can determine the acceleration by using the equation a = Δv / Δt. The y-intercept tells us the initial velocity at time t = 0.
Velocity is how fast an object is changing its position with respect to time. The x-axis is time and the y-axis is velocity in m/s. If velocity is positive, the object is moving away from the point of origin. If the velocity is negative, the object is moving towards the point of origin. We can also interpret velocity time graphs through the slope. When the slope is positive, the acceleration is positive, while when the slope is negative the acceleration is negative. Since acceleration is just the change in velocity over time, we can determine the acceleration by using the equation a = Δv / Δt. The y-intercept tells us the initial velocity at time t = 0.
Determining Velocity/Speed and Acceleration from a Position Time Graph
Velocity
The velocity or speed of an object is simply the slope of the position time graph. Velocity is how fast an object is moving over time. The velocity is simply the change in position divided by the change in time. While position is usually measured in m (meters), velocity is measured in m/s (meters per second) since it is the position over time. Instantaneous velocity is the velocity at a point or time t.
Acceleration
The acceleration is the change in the velocity. When velocity is changing at a constant rate, the acceleration is constant. The acceleration of a constant velocity is just Δv over time. This is usually measured in m/s^2. However, when the velocity is not changing at a constant rate, we would need to use the kinematic equation Δx = 1 / 2 * a * Δt^2 + Vi * Δt. By using the position graph, we know the values for Δx and Δt. Initial velocity can be determined by the slope of the position time graph at the initial position. If the object is not changing in position at t = 0, then the initial velocity is just 0. After plugging in the variables, we just solve for a, the acceleration.
Velocity
The velocity or speed of an object is simply the slope of the position time graph. Velocity is how fast an object is moving over time. The velocity is simply the change in position divided by the change in time. While position is usually measured in m (meters), velocity is measured in m/s (meters per second) since it is the position over time. Instantaneous velocity is the velocity at a point or time t.
Acceleration
The acceleration is the change in the velocity. When velocity is changing at a constant rate, the acceleration is constant. The acceleration of a constant velocity is just Δv over time. This is usually measured in m/s^2. However, when the velocity is not changing at a constant rate, we would need to use the kinematic equation Δx = 1 / 2 * a * Δt^2 + Vi * Δt. By using the position graph, we know the values for Δx and Δt. Initial velocity can be determined by the slope of the position time graph at the initial position. If the object is not changing in position at t = 0, then the initial velocity is just 0. After plugging in the variables, we just solve for a, the acceleration.
Determining Position, Distance and Displacement from a Velocity Time Graph
Position
The rule used when solving for position using a velocity time graph is that the change in position is the area under the curve when above the x-axis, and above the curve when below the x-axis.
Distance
The distance traveled is calculated by adding the area under the curve for positive velocity values and the area above the curve for negative velocity values. This is because distance is the total amount traveled, regardless of direction.
Displacement
Since displacement accounts for direction, you must subtract the area above the curve for the negative velocity values from the area under the curve for positive velocity values. Essentially, this allows us find the final position after we walk away from the point of origin and then walk towards the point of origin.
Connecting Representations of Motion
Slope of Position (Position Graph) -------> Velocity or the y-axis (Velocity Graph)
Slope of Velocity (Velocity Graph) ------> Acceleration or the y-axis (Acceleration Graph)
Strobe Diagram
A strobe diagram shows the position of an object at each second. The longer the distance between the dots, the faster an object is moving. This is because the object must have traveled a longer distance in the same amount of time. This is because as the ball rolls down the hill, it is accelerating. Therefore, it is picking up speed and the distance between the position of the object at each second begins to increase.
Problem Solving
Kinematic Equations
1. (Xf - Xi) =1 / 2 * a * t^2 + Vi * Δt
2. Vf = a * Δt + Vi
3. Vf^2 = Vi^2 + 2 * a * (Xf - Xi)
Kinematics Problem Solving revolves around using known variables to determine unknown variables. When we are given the context of a problem, we are also given some numbers for certain variables. We are given information regarding some of the following variables: Final Position (Xf), Initial Position (Xi), Final Velocity (Vf), Initial Velocity (Vi), Acceleration (a), Change in time (Δt). By plugging in the values for the variables we do know, we can solve for the unknown variables. It is also important to note that when an object is just starting to accelerate, the initial velocity is 0. Also, the acceleration for an object being pulled downward by gravity is -9.8m/s^2.
Kinematic Equations
1. (Xf - Xi) =1 / 2 * a * t^2 + Vi * Δt
2. Vf = a * Δt + Vi
3. Vf^2 = Vi^2 + 2 * a * (Xf - Xi)
Kinematics Problem Solving revolves around using known variables to determine unknown variables. When we are given the context of a problem, we are also given some numbers for certain variables. We are given information regarding some of the following variables: Final Position (Xf), Initial Position (Xi), Final Velocity (Vf), Initial Velocity (Vi), Acceleration (a), Change in time (Δt). By plugging in the values for the variables we do know, we can solve for the unknown variables. It is also important to note that when an object is just starting to accelerate, the initial velocity is 0. Also, the acceleration for an object being pulled downward by gravity is -9.8m/s^2.
Projectile Motion
A projectile is an object in the air that is not touching the air. A projectile is propelled upward by a initial velocity. The projectile reaches its peak or maximum height. At that point the vertical velocity is 0, and then the object begins to drop to the ground. The horizontal velocity is always constant. However, the vertical velocity follows the formula Δy = V0y * Δt - 1 / 2 * g * Δt^2 where g is the gravitation cosntant, and V0y is the initial vertical velocity.