Motion with constant acceleration Lecture deals with a very common type of motion: motion with constant acceleration After this lecture, you should know about: Kinematic equations Free fall. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 1

Summary of Concepts (from last lecture) kinematics: A description of motion position: your coordinates displacement: Δx = change of position distance: magnitude of displacement velocity: rate of change of position average : Δx/Δt instantaneous: slope of x vs. t speed: magnitude of velocity acceleration: rate of change of velocity average: Δv/Δt instantaneous: slope of v vs. t 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Motion with constant acceleration in 1D Kinematic equations An object moves with constant acceleration when the instantaneous acceleration at any point in a time interval is equal to the value of the average acceleration over the entire time interval. Choose t0=0: 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Motion with constant acceleration in 1D Kinematic equations (II) Because velocity changes uniformly with time, the average velocity in the time interval is the arithmetic average of the initial and final velocities: (1) (2) Putting (1) and (2) together: 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Motion with constant acceleration in 1D Kinematic equations (III) The area under the graph of velocity vs time for a given time interval is equal to the displacement Δx of the object in that time interval 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Motion with constant acceleration in 1D Kinematic equations (IV) Putting the following two formulas together another way: 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Motion with constant acceleration in 1D Kinematic equations (V) Δx = v0t + 1/2 at2 (parabolic) Δv = at (linear) v2 = v02 + 2a Δx (independent of time) 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Use of Kinematic Equations Gives displacement as a function of velocity and time Use when you don’t know or need the acceleration Shows velocity as a function of acceleration and time Use when you don’t know or need the displacement Gives displacement given time, velocity & acceleration Use when you don’t know or need the final velocity Gives velocity as a function of acceleration and displacement Use when you don’t know or need the time 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Example for motion with a=const in 1D: Free fall The Guinea and Feather tube Experimental observations: Earth’s gravity accelerates objects equally, regardless of their mass. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Free Fall Principles Objects moving under the influence of gravity only are in free fall Free fall does not depend on the object’s original motion Objects falling near earth’s surface due to gravity fall with constant acceleration, indicated by g g = 9.80 m/s2 g is always directed downward toward the center of the earth Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Summary: Constant Acceleration x = x0 + v0xt + 1/2 at2 vx = v0x + at vx2 = v0x2 + 2a(x - x0) Free Fall: (a = -g) y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2g(y - y0) x y up down Braking distance: When braking from 100 km/h to 0 km/h, one needs distance x. When braking from 200 km/h to 0 km/h, one needs 4x ! Because if one assumes that deceleration is constant, a, than the time it needs to get the velocity from 100 to zero is t and the time for 200 to 0 is 2 times t (linear dependence on t) while distance depends quadratically on t. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 1 A ball is thrown straight up in the air and returns to its initial position. For the time the ball is in the air, which of the following statements is true? 1 - Both average acceleration and average velocity are zero. 2 - Average acceleration is zero but average velocity is not zero. 3 - Average velocity is zero but average acceleration is not zero. 4 - Neither average acceleration nor average velocity are zero. correct Free fall: acceleration is constant (-g) Initial position = final position: Δx=0 averaged vel = Δx/ Δt = 0 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Free Fall dropping & throwing Initial velocity is zero Acceleration is always g = -9.80 m/s2 Throw Down Initial velocity is negative Throw Upward Initial velocity is positive Instantaneous velocity at maximum height is 0 vo= 0 (drop) vo< 0 (throw) a = g v = 0 a = g 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Throwing Down Question A ball is thrown downward (not dropped) from the top of a tower. After being released, its downward acceleration will be: 1. greater than g 2. exactly g 3. smaller than g 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 2 A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true: 1. velocity is zero and acceleration is zero 2. velocity is not zero and acceleration is zero 3. velocity is zero and acceleration is not zero 4. velocity is not zero and acceleration is not zero correct Acceleration is the change in velocity. Just because the velocity is zero does not mean that it is not changing. At the top of the path, the velocity of the ball is zero, but the acceleration is not zero. The velocity at the top is changing, and the acceleration is the rate at which velocity changes. Acceleration is not zero since it is due to gravity and is always a downward-pointing vector. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 3A Dennis and Carmen are standing on the edge of a cliff. Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball is moving fastest when it hits the ground? 1. Dennis' ball 2. Carmen's ball 3. Same v0 Dennis Carmen H vA vB Correct: v2 = v02 -2gΔy On the dotted line: Δy=0 ==> v2 = v02 v = ±v0 When Dennis’s ball returns to dotted line its v = -v0 Same as Carmen’s 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 3B Dennis and Carmen are standing on the edge of a cliff. Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball hits the ground at the base of the cliff first? 1. Dennis' ball 2. Carmen's ball 3. Same correct Time for Dennis’s ball to return to the dotted line: v = v0 - g t v = -v0 t = 2 v0 / g This is the extra time taken by Dennis’s ball v0 Dennis Carmen y=y0 vA vB y=0 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 4 An object is dropped from rest. If it falls a distance D in time t then how far will if fall in a time 2t ? 1. D/4 2. D/2 3. D 4. 2D 5. 4D Correct x=1/2 at2 Follow-up question: If the object has speed v at time t then what is the speed at time 2t ? 1. v/4 2. v/2 3. v 4. 2v 5. 4v Correct v=at 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 5 Which of the following statements is most nearly correct? 1 - A car travels around a circular track with constant velocity. 2 - A car travels around a circular track with constant speed. 3- Both statements are equally correct. correct The direction of the velocity changes when going around circle. Speed is the magnitude of velocity -- it does not have a direction and therefore does not change 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Motion in 2D After this lecture, you should know about: Vectors. Displacement, velocity and acceleration in 2D. Projectile motion. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 1

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University One Dimension } Define origin Define sense of direction Position is a signed number (direction and magnitude) Displacement, velocity, acceleration are also specified by signed numbers Reference Frame …-4 -3 -2 -1 1 2 3 4… 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Vectors There are quantities in physics which are determined uniquely by one number: Mass is one of them. Temperature is one of them. Speed is one of them. We call those scalars. There are others where you need more than one number; for instance for 1D motion, velocity has a certain magnitude-- that's the speed-- but you also have to know whether it goes this way or that. So there has to be a direction. We call those vectors. 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Two Dimensions Again, select an origin Draw two mutually perpendicular lines meeting at the origin Select +/- directions for horizontal (x) and vertical (y) axes Any position in the plane is given by two signed numbers A vector points to this position 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Properties of vectors Equality of two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Negative Vectors One vector is the negative of another one if they have both the same magnitude but are 180° apart (opposite directions) Resultant Vector The resultant vector is the sum of a given set of vectors Position can be anywhere in the plane 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Adding and subtracting vectors geometrically R=R1+R2 D=R2-R1 y R1 R2 D x 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Components of a Vector A component is a part It is useful to use rectangular components These are the projections of the vector along the x- and y-axes The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis Then, one can define the component vectors Attention: θ is measured counter-clock-wise with respect to the positive x-axis 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Components of a vector (II) The components are the legs of the right triangle whose hypotenuse is May still have to find θ with respect to the positive x-axis 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Adding Vectors Algebraically Choose a coordinate system and sketch the vectors Find the x- and y-components of all the vectors Add all the x-components This gives Rx: Add all the y-components This gives Ry: Use the Pythagorean Theorem to find the magnitude of the resultant: Use the inverse tangent function to find the direction of R: Inversion is not unique, the value will be correct only if the angle lies in the first or fourth quadrant In the second or third quadrant, add 180° 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 6 Can a vector have a component bigger than its magnitude? Yes No The square of magnitude of a vector is given in terms of its components by R2= Rx 2+ Ry 2 Since the square is always positive the components cannot be larger than the magnitude 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Example 7 The sum of the two components of a non-zero 2-D vector is zero. Which of these directions is the vector pointing in? 45o 90o 135o 180o 135o -45o The sum of components is zero implies Rx = - Ry The angle, θ = tan-1(Ry / Rx) = tan-1 -1 = 135o = -45o (not unique, ± multiples of 2 θ) 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

2D motion: Displacement The position of an object is described by its position vector, The displacement of the object is defined as the change in its position 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

2D motion: Velocity and acceleration The average velocity is the ratio of the displacement to the time interval 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 The average acceleration is defined as the rate at which the velocity changes The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero Ways an object might accelerate: The magnitude of the velocity (the speed) can change The direction of the velocity can change Both the magnitude and the direction can change 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Kinematics in Two Dimensions x = x0 + v0xt + 1/2 axt2 vx = v0x + axt vx2 = v0x2 + 2ax Δx y = y0 + v0yt + 1/2 ayt2 vy = v0y + ayt vy2 = v0y2 + 2ay Δy x and y motions are independent! They share a common time t 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

2D motion: Projectile motion Dimensional Analysis: Motion of a soccer ball Strategy: 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Kinematics for Projectile Motion ax = 0 ay = -g y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2g Δy x = x0 + vxt vx = v0x x and y motions are independent! They share a common time t 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Projectile Motion y ~ -x2, i.e. parabolic dependence on x 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Projectile Motion: Maximum height reached Time taken for getting there 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Projectile Motion: Maximum Range 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Projectile Motion at Various Initial Angles Complementary values of the initial angle result in the same range The heights will be different The maximum range occurs at a projection angle of 45o 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Soccer Ball Make sense of what you get Check limiting cases 4/24/2017 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University