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PHYS 218 sec. 517-520 Review Chap. 3 Motion in 2 or 3 dimensions

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Position & velocity vectors 1-dim 2-dim

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Components of acceleration We focus on 2-dim. motions. Parallel part This is similar to the case of 1-dim. motion. The magnitude of the velocity can change and it depends on the sign of the acceleration. But the direction of the velocity does not change as the acceleration is always along the velocity.

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Perpendicular part The magnitude of the velocity does not change. But the direction of the velocity changes.

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When speed is constant along a curved path. Acceleration is normal to the path. Since the speed doesn’t change, there is no acceleration parallel to the velocity. Change of velocity at point P When speed is increasing/decreasing along a curved path.

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Projectile motion A projectile: any object that is given an initial velocity and then follows a path determined entirely by the gravitational acceleration. (We ignore air resistance.) trajectory We choose the Cartesian (x-y) coordinate system to describe this motion, since the acceleration is always in the negative y-direction. You can easily find that, in general, the acceleration and velocity are not parallel nor perpendicular to each other. So the acceleration changes the direction of the motion as well as its speed.

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Projectile motion (equations) x-direction y-direction These are equations of motion for a projectile. This completely describes the projectile motion and most problems can be solved using these equations. Of course, these equations are valid only when air resistance is neglected and the gravitational acceleration is a constant.

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Projectile motion (application) Maximum height h

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Range R R is determined by the point where the projectile hits the ground again.

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Different initial and final heights Express R with initial variables

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Circular motion This is another important example of a 2-dim. motion. The direction of the velocity is changing. This means that the acceleration must have a component which is perpendicular to the velocity even if the speed is constant. Cf. In the projectile motion, the direction of the acceleration is always fixed. Uniform circular motion

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Uniform Circular Motion (direction of the acceleration) The acceleration is always pointing to the center of the circular path. counterclockwise clockwise It is called a centripetal acceleration.

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Uniform Circular Motion (acceleration) These two triangles are similar! Explained in the class

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Uniform Circular Motion (acceleration) Period (T): the time for one revolution, i.e. the time for the object to complete one trip around the circle. In a time T, the particle travels a distance equal to the circumference, 2 R. Since this motion has a uniform speed, its average speed is the instantaneous speed.

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Nonuniform Circular Motion This is a circular motion with varying speed. Note Tangential acceleration, zero for uniform circular motion

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Relative velocity Relative velocity: velocity seen by a particular observer. A B C When two observers B and C measure the velocity of A, they have different results as B and C are moving relative to each other. Frame of reference: Each observer forms a reference frame, i.e. coordinate system and time scale. Observer B measures the velocity of A by its own ruler and watch. Same is true for the observer C. all physical quantities are measured in that reference frame.

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Origin of A’s reference frame Origin of B’s reference frame x P/A : position of P measured by A x P/B : position of P measured by B x B/A : position of B measured by A x A/B : position of A measured by B Galilean velocity transformation

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This is what we observe in daily life. Not valid if the velocity is large, i.e. close to the speed of light. Einstein’s special theory of relativity Two postulates (consider two reference frames moving with a constant speed relative to each other) The Principle of Relativity - The laws of physics are same in different reference frames. The Principle of Invariant Light Speed – The speed of light is the same regardless of the reference frame. Even if P is at rest in the reference frame B, P is moving in the reference frame A! Absolute rest frame? No Yes Galilean relativity

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Motion in Two Dimensions

Motion in Two Dimensions

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