velocity and position by integration. non-constant acceleration example
TRANSCRIPT
Velocity and Position by Integration
• As we saw while deriving the kinematic equations for constant acceleration, we can express velocity as the integral of acceleration and position as the integral of velocity.
• In this case, we will not assume that is a constant.
• Likewise for position and velocity,
Non-constant Acceleration Example
Chapter 2 SummaryMotion Along a Straight Line
• Velocity
• Average:
• Instantaneous:
• Slope of position vs. time
• Acceleration
• Average:
• Instantaneous:
• Slope of velocity vs. time; Curvature of position vs. time
Chapter 2 SummaryMotion Along a Straight Line
• Kinematic equations for motion with constant acceleration
• Free fall acceleration due to gravity:
• Motion with varying acceleration
Chapter 3 OutlineMotion in Two or Three Dimensions
• Position and velocity vectors
• Acceleration vectors
• Parallel and perpendicular components
• Projectile motion
• Uniform circular motion
• Relative velocity
Position Vector
• The position vector points from the origin to point , the position of the object.
• We can express the vector in terms of its , , and components.
• The position unit vector gives the direction from the origin to the object.
Velocity Vector
• The velocity vector is found from the time derivative of the position vector.
• The velocity is tangent to the path at each point.
• In component form:
Velocity Vector
• The velocity in each direction is just the time derivative of the coordinate of that direction.
• The magnitude of the velocity (speed) is given by:
Acceleration Vector
• The acceleration vector is found from the time derivative of the velocity vector.
• While we might typically think of acceleration as a change in speed, it is very important that we understand that it is a change in velocity.
• As we will discuss later in this chapter, in uniform circular motion, the speed is not changing, but the direction, and therefore velocity is constantly changing.
Acceleration Vector
• In component form:
• Or,
Parallel and PerpendicularComponents of Acceleration
• We can resolve the acceleration into its components parallel to the velocity (along the path) and perpendicular to the velocity.
• The parallel component, only changes the magnitude of the velocity, its speed.
• The perpendicular component, only changes the direction of the velocity, so its speed remains constant.
Projectile Motion
• Any body that is given an initial velocity and follows a path determined solely by the effects of gravity and air resistance is a projectile.
• The path the projectile follows is its trajectory.
• Initially, we will consider the simplest model in which we neglect the effects of air resistance, and the curvature of the earth.
Projectile Motion
• While a projectile moves in three-dimensional space, we can always reduce the problem to two dimensions by choosing to work in the vertical - plane that contains the initial velocity.
• We can simplify this further by treating the and components separately.
• The vertical and horizontal motions are independent
Projectile Motion
• In the ideal model, we only consider the force due to gravity, so there is no acceleration in the direction.
• In the direction, we have an acceleration due to gravity of downward. For the following equations, we will use a coordinate system in which up is positive.
• While not required, it is often simplest to set the origin as the initial position of the projectile, so that at .
Projectile Motion
• We can represent the initial velocity in terms of its components to rewrite the equations for position and velocity.
• We have also taken the initial position to be at the origin.
Trajectory Shape
• The previous equations tell us the position and velocity at each time, but to see the shape of the trajectory, we need to look at the vertical position as a function of horizontal position. ()
• Note that is a function of . This gives rise to a parabola.
Projectile Motion Example #1
Projectile Motion Example #2