Example problem: Free Fall

Transcription

1 Example problem: Free Fall A ball is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The ball just misses the edge of the roof on its way down. Determine A. the time needed for the ball to reach its maximum height. B. the maximum height itself. C. the time needed for the ball to return to the height from which it was thrown and the velocity of the ball at that instant. D. the time needed for the ball to reach the ground. E. the velocity and position of the ball at t = 5s. Neglect air drag. Some helpful equations:

2 Today s lecture Vectors: What is a vector? Why do we need vectors in physics? Vector properties. How do we add/subtract 2 vectors? How do we determine the components of a vector?

3 Chapter 3: Vectors and Two-dimensional motion

4 There are two fundamentally different types of physical/mathematical quantities: 1) Vectors 2) Scalars A scalar quantity is fully characterized by its magnitude. A vector must be characterized by magnitude and direction. It is represented by an arrow. What is a vector?

5 Why do we need vectors in physics? Until now we only talked about objects moving in 1 dimension. Due to this limitation it was sufficient to say that an object moves e.g. 10 m. The direction was obvious, since there was only 1 direction (no vector needed). This was, however, a strong limitation! In reality objects can move in more than one direction. In principle, objects can move in 3 dimensions in space. In this chapter, we restrict ourselves to motion in two dimensions. In a two dimensional coordinate system, we can identify an object s position by a vector pointing from the origin to its position. This is called the position vector.

6 How do we describe vectors? - Notation! A vector is an arrow. Thus, we use the following notation: A vector can be fully described in two ways: 1. By providing its x- and y-components: 2. By providing its magnitude (length), R, and its angle, θ, with respect to the x-axis (counterclockwise).

7 Equality of two vectors Two vectors are equal, if they have the same magnitude and the same direction. Any vector can be moved parallel to itself without being affected. AB = DC A D AD = BC B C

8 Negative vectors The negative vector of is denoted by. It is a vector having equal magnitude, but opposite direction. A A B B

9 Multiplication of a vector with a number The product of a vector and a scalar, k, is a vector. The magnitude of the resulting vector is k times the magnitude of the original vector. This operation is called scalar multiplication. Example (k = 3): How do we calculate this? - We multiple every component of the vector by k.

10 How do we add two vectors? There are 2 methods to add two vectors, and : 1. Geometrical method: Draw the first vector with its tail located at the origin of a coordinate system. Draw the second vector with its tail located at the head of the first vector. Connect the tail of the first with the head of the second vector to get the sum. 2. Algebraic method: Add the x/y-components of both vectors to get the x/y-component of the resulting vector:

11 Again there are 2 methods: 1. Geometrical method: How do we subtract two vectors? Remember: The negative of a vector has the same magnitude as its original, but points into the opposite direction. Add the negative of the vector that you plan to subtract. 2. Algebraic method: Subtract the x/y-components of both vectors to get the x/y-component of the resulting vector:

12 How do we determine the components of a vector? This plots shows the original vector and its component-vectors (projections along the x- and y-axes). The magnitudes of the component vectors are the components of the original vector. These 3 vectors form a rectangular triangle. The angle, θ, is measured with respect to the x-axis (counterclockwise). Pythagoras:

13 Summary A vector is an arrow characterized by its magnitude and direction. Magnitude: Direction: Vectors are required to describe the position of objects and their motion in more than one direction. Two vectors are identical, if their magnitudes and directions are the same. Multiplying a vector with a scalar, k, changes its magnitude by a factor of k: Two vectors can be added geometrically or algebraically: A minus sign reverses a vector s direction. Subtracting 2 vectors is the same as adding a negative vector. A vector s components can be determined:

14 Displacement, Velocity, Acceleration in 2d In 2d problems, the position of an object is determined by its position vector. If an object moves from an initial to a final position, the displacement will be a vector, too (unit: m): Velocity (unit: m/s) and acceleration (unit: m/s 2 ) are also vectors: An object can accelerate in different ways: 1. The magnitude of the velocity changes with time (the same as 1d problems) 2. The direction of the velocity may change with time, e.g. circular motion, at constant speed. 3. Magnitude and direction change in parallel.

15 Projectile motion A projectile is any body, given an initial velocity, that then follows a path determined by the effects of gravity and air resistance. We neglect air resistance.

16 The physics of Wiley E Coyote

17 How would Coyote E. Wiley fall down a cliff in reality? As Mr. Coyote runs off the cliff, he has horizontal velocity. A change in velocity is acceleration, in this case horizontal acceleration, which must come from a force in the horizontal direction. vfx=vox+axt If we ignore air resistance (horizontal force = 0), then there is no horizontal force to slow him down horizontally. vfx=vox Thus, Mr. Coyote will travel horizontally at the same speed the whole time until he hits the ground!

18 How would Coyote E. Wiley fall down a cliff in reality? Vertical motion is treated separately. As soon as the coyote leaves the cliff he will experience a vertical force due to gravity. This force will cause him to start to accelerate in the vertical direction. As he falls he will be going faster and faster in the vertical direction. The horizontal and vertical components of the motion of an object going off a cliff are separate from each other, and can not affect each other.

19 Mathematical description of projectile motion v! o θ o intial velocity vector initial direction of velocity vector vy = 0 at top of trajectory vx = vxo remains the same throughout trajectory because there is no acceleration along the x-direction. We can use the equations to describe 1d motion with constant acceleration for the horizontal and vertical direction separately: