Honors Physics “Mechanics for Physicists and Engineers”Agenda for Today • Advice • 1-D Kinematics • Average & instantaneous velocity and acceleration • Motion with constant acceleration • Freefall
Kinematics Objectives • Define average and instantaneous velocity • Caluclate kinematic quantities using equations • interpret and plot position -time graphs • be able to determine and describe the meaning of the slope of a position-time graph
Kinematics • Location and motion of objects is described using Kinematic Variables: • Some examples of kinematic variables. • position r vector, (d,x,y,z) • velocity v vector • acceleration a vector • Kinematic Variables: • Measured with respect to a reference frame. (x-y axis) • Measured using coordinates (having units). • Many kinematic variables are Vectors, which means they have a direction as well as a magnitude. • Vectors denoted by boldface Vor arrow above the variable
Motion • Position: Separation between an object and a reference point (Just a point) • Distance: Separation between two objects • Displacement of an object is the distance between it’s final position df and it’s initial position d i (d f - di)= d • Scalar: Quantity that can be described by a magnitude(strength) only • Distance, temperature, pressure etc.. • Vector: A quantity that can be described by both a magnitude and direction • Force, displacement, torque etc.
Speed and Velocity • Speed describes the rate at which an object moves. Distance traveled per unit of time. • Velocity describes an objects’ speed and direction. • Approximate units of speed
Motion in 1 dimension • In general, position at time t1 is usually denoted d, r(t1) or x(t1) • In 1-D, we usually write position as x(t1 ) but for this level we’ll use d • Since it’s in 1-D, all we need to indicate direction is + or . • Displacement in a time t = t2 - t1isx = x2 - x1= d2 -d1 x some particle’s trajectoryin 1-D x2 x x1 t1 t2 t t
1-D kinematics • Velocity v is the “rate of change of position” • Average velocity vav in the time t = t2 - t1is: x trajectory d2 x Vav = slope of line connecting x1 and x2. d1 t1 t2 t t
1-D kinematics... • Instantaneous velocity v is definedas the velocity at an instant of time (t= 0) • Slope formula becomes undefined at t = 0 x soV(t2 ) = slope of line tangent to path at t2. x2 x x1 Calculus Notation t1 t2 t t
v 60 t 1 2 More 1-D kinematics • We saw that v = x / t • so therefore x = v t( i.e. 60 mi/hr x 2 hr = 120 mi ) • See text: 3.2 • In “calculus” language we would write dx = v dt, which we can integrate to obtain: • Graphically, this is adding up lots of small rectangles: v(t) + +...+ = displacement t
1-D kinematics... • Acceleration a is the “rate of change of velocity” • Average acceleration aav in the time t = t2 - t1is: • Andinstantaneous acceleration a is definedas:The acceleration when t = 0 . Same problem as instantaneous velocity. Slope equals line tangent to path of velocity vs time graph.
Problem Solving • Read ! • Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information in given, what is asked for, and the meaning of all the terms used in stating the problem. • Watch your units ! • Always check the units of your answer, and carry the units along with your numbers during the calculation. • Understand the limits ! • Many equations we use are special cases of more general laws. Understanding how they are derived will help you recognize their limitations (for example, constant acceleration).
IV. Displacement during acceleration. • You accelerate from 0 m/s to 30 m/s in 3 seconds, how far did you travel? • What if a car initially at 10 m/s, accelerates at a rate of 5 m/s2 for 7 seconds. How far does it move? • df=1/2at2 + vit + di • C. An airplane must reach a speed of 71 m/s for a successful takeoff. What must be the rate of acceleration if the runway is 1.0 km long? • d = (vf2 - vi2) /2a
Recap • If the position x is known as a function of time, then we can find both velocity vand acceleration a as a function of time! x t v t a t
Recap • So for constant acceleration we find: x t v • From which we can derive: t a t
IV. Acceleration due to gravity • The acceleration of a freely falling object is 9.8 m/s2 (32 ft/s2) towards the earth. • The farther away from the earth’s center, the smaller the value of the acceleration due to gravity. For activities near the surface of the earth (within 5-6 km or more) we will assume g=9.8 m/s2 (10 m/s2). • Neglecting air resistance, an object has the same acceleration on the way up as it does on the way down. • Use the same equations of motion but substitute the value of ‘g’ for acceleration ‘a’.
Recap of kinematics lectures • Measurement and Units (Chapter 1) • Systems of units • Converting between systems of units • Dimensional Analysis • 1-D Kinematics • Average & instantaneous velocity and and acceleration • Motion with constant acceleration