Describing motion in a straight line Physics 221

Learning Goals for Chapter 2 Looking forward at … how the ideas of displacement and average velocity help us describe straight-line motion. the meaning of instantaneous velocity; the difference between velocity and speed. how to use average acceleration and instantaneous acceleration to describe changes in velocity. how to solve problems in which an object is falling freely under the influence of gravity alone. how to analyze straight-line motion when the acceleration is not constant.

Displacement Displacement is defined as the change in position, x: Displacement can be + or – depending on which direction the position has changed. Usually we take a move to the right as positive and a move to the left as negative. Technically, this is not a displacement, since displacement is a vector quantity. The quantity shown above is a distance, which is a scalar quantity. However, in a one-dimensional system, the sign of the distance will give a direction.

Displacement, time, and average velocity A particle moving along the x-axis has a coordinate x. The change in the particle’s coordinate is The average x-velocity of the particle is Technically, this is not a velocity, since velocity is a vector quantity. The quantity shown above is a speed, which is a scalar quantity. However, in a one-dimensional system, the sign of the speed will give a direction.

Average Speed Average speed is defined as Example: A car travels at an average speed of 55 mph for 1.3 hours and then 65 mph for 0.73 hours. What is the total distance travelled by the car and the average speed for the trip? Find the distance traveled for each part of the trip then add those numbers to get the total distance traveled. Then you will be able to get the average speed since the total time required is 2.03 hr. 5

Rules for the sign of x-velocity

Average velocity The winner of a 50-m swimming race is the swimmer whose average velocity has the greatest magnitude. That is, the swimmer who traverses a displacement Δx of 50 m in the shortest elapsed time Δt.

Average Velocity Average velocity is defined as Average velocity is the change in position of an object with time. It can be + or –. Usually we take the average velocity as positive when the object moves to the right and negative when it moves to the left.

A position-time graph Graphs are a standard method of representation for physical quantities; note the title, axes labels and tick marks.

Instantaneous velocity The instantaneous velocity is the velocity at a specific instant of time or specific point along the path and is given by vx = dx/dt. The average speed is not the magnitude of the average velocity!

Finding velocity on an x-t graph

Finding velocity on an x-t graph

Finding velocity on an x-t graph

x-t graphs

Motion diagrams Here is a motion diagram of the particle in the previous x-t graph.

He described acceleration, which is the rate of change of velocity. Galileo Galilei (1564 – 1642) was the first person to analyze motion in terms of measurements and mathematics. He described acceleration, which is the rate of change of velocity. Acceleration has units of m/s2 so an acceleration of 10 m/s2 means the velocity is changing by 10 m/s every second. Galileo Galilei is so famous we know him by his first name only. Galileo. Galileo, age 60, drawn by contemporary Ottavio Leoni in 1624. 16

Average acceleration Acceleration describes the rate of change of velocity with time. The average x-acceleration is “Acceleration” is used to describe both the vector and scalar quantity, which may be confusing in some cases. Fortunately, the sign of the acceleration is used to indicate directionality.

Average Acceleration Average acceleration is defined as It is the change in velocity with time. A velocity can change when the speed changes and/or the direction changes. Average acceleration can be + or -. Again, it is + if it points to the right and – if it points to the left.

Instantaneous acceleration The instantaneous acceleration is ax = dvx/dt.

Signs of velocity and acceleration A car speeds up while moving to the right. Which way does the acceleration point? What are the signs of the velocity and acceleration? +, + A car slows down while moving to the right. Which way does the acceleration point? What are the signs of the velocity and acceleration? +, – A car speeds up while moving to the left. Which way does the acceleration point? What are the signs of the velocity and acceleration? –, – A car slows down while moving to the left. Which way does the acceleration point? What are the signs of the velocity and acceleration? –, +

Air Force Doctor John Stapp testing human limits from acceleration Air Force Doctor John Stapp testing human limits from acceleration. He determined that humans could withstand accelerations of about 450 m/s2 for brief periods without suffering major injuries or death. In 1954 Stapp hit the speed of 632 mph in a rocket sled and the sled was brought to a stop in less than a second. He is eyes were bleeding because every blood vessel had burst. He suffered lingering eye issues from some blood vessels being permanently damaged. His work eventually led to designs for pilot restraint systems, ejection seats and seatbelts in cars. http://www.youtube.com/watch?v=s4tuvOer_GI

Finding acceleration on a vx-t graph

A vx-t graph

Motion diagrams Here is the motion diagram for the particle in the previous vx-t graph.

A vx-t graph

Motion diagrams Here is the motion diagram for the particle in the previous vx-t graph.

Motion with constant acceleration

Motion with constant acceleration

A position-time graph

Velocity from Graphical Position ay>0 ay<0

Instantaneous Velocity Observation: Any smooth curve becomes linear at a sufficiently high magnification. Instantaneous velocity is the local slope of the curve.

Example A particle has position x(t) = (3t - t3) where x is in m and t is in s. What is the particle’s position and velocity at t=2 s? x = [3(2) – (23)] = 6 m – 8 m = -2 m v = dx/dt = (3 - 3t2) = [3 - 3(22)] = 3 m/s – 12 m/s = – 9 m/s Plot position and velocity for -3 s < t < 3 s. Draw a diagram to illustrate the motion.

The equations of motion with constant acceleration The four equations below apply to any straight-line motion with constant acceleration ax.

Freely falling bodies Free fall is the motion of an object under the influence of only gravity. In the figure, a strobe light flashes with equal time intervals between flashes. The velocity change is the same in each time interval, so the acceleration is constant.

A freely falling coin If there is no air resistance, the downward acceleration of any freely falling object is g = 9.8 m/s2 = 32 ft/s2.

Up-and-down motion in free fall Position as a function of time for a ball thrown upward with an initial speed of 15.0 m/s.

Up-and-down motion in free fall Velocity as a function of time for a ball thrown upward with an initial speed of 15.0 m/s. The vertical velocity, but not the acceleration, is zero at the highest point.

Velocity and position by integration The acceleration of a car is not always constant. The motion may be integrated over many small time intervals to give and

Finding Position from Velocity or the change in position Ds = sf – si = the area under the velocity graph, which is the integral of the velocity over the time interval

Drag Racer’s Displacement The figure shows the velocity of a drag racer. How far does the racer move during the first 3.0 s? Solution: The net distance traveled is the area under the velocity curve shown in blue. This is a triangle with sides 12 m/s and 3.0 s. The area of this triangle is: A = ½(12 m/s)(3 s) = 18 m which equals the change in position during this time interval. Thus, the drag racer moves 18 m in the first 3 seconds.

Drag Racer’s Position Find an algebraic expression for the drag racer’s position from the previous slide. Assume that si = 0 m and ti = 0 s. (b) Plot a position vs. time graph. Solution: (a) The speed of the racer increases linearly with t, with v(t) = 4 t m/s. The position is: s(t) = si + 0∫t v(t1) dt1 = 0 + 0∫t 4 t1dt1 = [2 t12]0t = 2 t2 m (b) The graph is a parabola from the origin, as shown.