Chapter 6 Motion In Two-Dimensional

Motion in Two Dimensions Using ________signs is not always sufficient to fully describe motion in more than one dimension. ______ can be used to more fully describe motion. Still interested in displacement, velocity, and acceleration. We can describe each of these in terms of vectors.

Displacement The _______ of an object is described by its position vector,. The displacement of the object is defined as the change in its position.

Velocity The average velocity is the ratio of the ______________ to the ___________ for the displacement. The instantaneous velocity is the limit of the average velocity as Δt approaches zero. The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion.

Acceleration The average acceleration is defined as the rate at which the ______ changes. The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero.

Unit Summary (SI) Displacement m (All displacement has the dimensions of length, L) Average velocity and instantaneous velocity m/s (All velocities have the dimensions of L/T) Average acceleration and instantaneous acceleration m/s 2 (All accelerations have the dimensions of L/T 2.

Ways an Object Might Accelerate The ____________of the velocity (the speed) can change The ______________ of the velocity can change. Even though the magnitude is constant _____the ________and the ________ can change.

Projectile Motion An object may __________x and y directions ___________. It moves in two dimensions The form of two dimensional motion we will deal with is an important special case called _____________.

Assumptions of Projectile Motion We may ignore ___________. We may ignore the ___________________ Disregard the _________ of the earth. With these assumptions, an object in projectile motion will follow a parabolic path.

Characteristics of Projectile Motion The x- and y-directions of motion are completely _____________ of each other. The x-direction is _____________: a x = 0 The y-direction is ____________: a y = -g The _____________can be broken down into its ____________________:

Projectile Motion

Some Details About the Rules x-direction: the horizontal motion a x = 0 This is constant throughout the entire motion. ∆x = v o x t This is the only operative equation in the x- direction since there is uniform velocity in that direction. The _______________is dependent on the _______________and the _____________.

More Details About the Rules y-direction: the vertical motion V y = 0 m/s (at height) Free fall problem a = -g Take the positive direction as upward. Uniformly accelerated motion, so use the ____________________to solve for this motion.

Velocity of the Projectile The _________ of the projectile at any point of its motion is the _________of its x- and y-components at that point: Remember to be careful about the angle’s quadrant.

Projectile Motion Summary Provided air resistance is negligible, the horizontal component of the velocity remains constant. Since a x = 0 The vertical component of the velocity v y is equal to the free fall acceleration –g The acceleration at the top of the trajectory is not zero.

Projectile Motion Summary (cont.) The vertical component of the velocity v y and the displacement in the y-direction are identical to those of a freely falling body. Projectile motion can be described as a superposition of two independent motions in the x- and y-directions. The projection angle (θ) that the velocity vector makes with the x-axis is given by: Θ = tan -1 v y /v x

Problem-Solving Strategy Select a coordinate system and sketch the path of the projectile Include initial and final positions, velocities, and accelerations Resolve the initial velocity into x- and y-components Treat the horizontal and vertical motions independently

Problem-Solving Strategy (cont.) Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile. Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile.

Sample Problem A long jumper leaves the ground at an angle of 20.0 o to the horizontal and at a speed of 11.0 m/s. (a) How far does he jump? (Assume that his motion is equivalent to that of a particle.) (b) What maximum height is reached?

Circular Motion Specific type of 2- dimensional motion. Uniform circular motion: The movement of an object or particle trajectory at a constant speed around a circle with a fixed radius.

Displacement During Circular Motion As an object moves around a circle its ___________does not change lengths but its _________ does. To get the object’s velocity we need the change in displacement.

Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration. The ____________ ____________ is due to the change in the ___________ of the velocity.

Centripetal Acceleration Centripetal refers to “center- seeking.” The direction of the velocity changes. The acceleration is directed toward the ______ of the circle of motion.

Centripetal Acceleration The magnitude of the centripetal acceleration is given by: This direction is toward the center of the circle. So, whenever an object is moving in a circular motion it has centripetal acceleration. Force causing this acceleration will be discussed later.

Forces Causing Centripetal Acceleration There are two things happening as an object moves along a circular path: Object wants to move along a straight line. Object wants to move along the circular path. Which one wins out?

Forces Causing Centripetal Acceleration Newton’s Second Law says that the centripetal acceleration is accompanied by a force. F C = ma C F C stands for any force that keeps an object following a circular path. Tension in a string Gravity Force of friction

Centripetal Force General equation: If the force ___________, the object will move in a ____________tangent to the circle of motion. Centripetal force is a classification that includes forces acting toward a central point. It is not a force in itself.

Centripetal Force In each example identify the centripetal force.

Sample Problem A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the stopper.

Chapter 6 Motion In Two-Dimensional THE END