velocity and position by integration. non-constant acceleration example

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