Distance and Displacement

Frames of Reference Whenever you describe something that is moving, you are comparing it with something that is assumed to be stationary, or not moving Frame of reference – a system of objects that are not moving with respect to one another

Frames of Reference How fast is the earthearth moving? There is no single frame of reference that is always correct. North pole – 1000 mph Sun – 67,000 mph Center Milky Way – 560,000 mph Other Galaxies – 671,000 mph

How fast is the earth moving? 1.3 million miles per hour

Relative Motion Relative motion – movement in relation to a frame of reference Which frame should you choose? The one that is most meaningful for the situation

Distance Distance – the length of a path between two points The entire length of the path is measured, regardless of its twists and turns. A number and a unit must be used. The SI unit for distance is the meter (m), but you should use units that are appropriate for the distance being measured…cm, m, km, etc. Ex. – 35 m

Displacement Displacement – the direction from the starting point and the length of a straight line from the starting point to the ending point A number, unit, and direction must be included. Ex. – 300 m North Taking a round trip and ending up back at the starting point results in a displacement of zero.

Scalars and Vectors A scalar is a quantity with magnitude (size) but no direction. A vector is a quantity that has magnitude and direction. Is distance a scalar or a vector? Scalar Is displacement a scalar or a vector? Vector

Combining Displacements Displacements are combined using vector addition. A vector is often represented by an arrow. The length of the arrow indicates the magnitude. The direction of the arrow indicates the direction.

Combining Displacements Vector addition (head-to-tail method): The tail of the first arrow is placed a the starting point. The tail of the second arrow is placed at the head of the first arrow. The tail of the third arrow is placed at the head of the second arrow…Etc. The resultant vector (the sum of two or more vectors) can be drawn from the tail of the first arrow to the head of the last arrow.

Along a straight line… If two displacements, represented by two vectors (arrows), are in the same direction… you can add their magnitudes. If two displacements are in opposite directions… you can subtract their magnitudes. The resulting displacement will be in the direction of the largest vector. Don’t forget to include a direction along with a number and unit.

NOT along a straight line… Displacement vectors in different directions can be combined by graphing. The length of the resultant vector can be determined using the Pythagorean theorem… a 2 + b 2 = c 2. The a and b terms represent the short sides of the triangle, while the c term represents the hypotenuse. The resultant vector will be the hypotenuse.

Speed and Velocity

Speed Speed – the ratio of distance to time (in other words… distance divided by time) The SI unit for speed is meters per second (m/s), but you should use units that are appropriate for the speed being measured…cm/s, m/s, km/hr, etc. Is speed a scalar or a vector? Scalar

Average vs. Instantaneous Speed Average speed ( v )is computed for the entire duration of a trip. Average speed = total distance / total time v = d / t Instantaneous speed (v) is measured at a particular instant. Instantaneous speed can be measured by a car’s speedometer (or a cop’s radar).

Graphing Motion Time is the independent variable, graphed on the x- axis (the horizontal axis). Distance is the dependent variable, graphed on the y- axis (the vertical axis). Slope= rise over run = change in y-axis over change in x-axis = change in distance over change in time = speed

Slope (of a dist-time graph) Since slope = speed… A steep slope indicates a fast speed. A shallow slope indicates a slow speed. A horizontal line indicates a speed of zero. (The object is standing still.) A curving line with an increasing slope represents positive acceleration (speeding up). A curving line with a decreasing slope represents negative acceleration (slowing down).

Velocity Velocity – the speed and direction in which an object is moving Is velocity a scalar or a vector? Vector A change in velocity could be… A change in speed A change in direction Both Velocity can change even if speed doesn’t. Ex. - A marble going at a constant speed around a circular track is continually changing its velocity because its direction is continually changing.

Combining Velocities Two or more velocities can be added by vector addition (the head-to-tail method used for combining displacements). Add velocities in the same direction. Subtract velocities in opposite directions. Graph velocities in varying directions and use the Pythagorean theorem as needed.

Acceleration

Acceleration – the rate at which velocity changes Acceleration can be… A change in speed A change in direction Both Is acceleration a scalar or a vector? Vector

Changes in Speed Positive acceleration speed increases pressing on the gas pedal or free fall Negative acceleration (aka deceleration) speed decreases applying the brakes Zero acceleration speed is constant using the cruise control

Free Fall Free fall – the movement of an object toward Earth solely because of gravity Acceleration due to gravity (g) = 9.8 m/s 2 This means that the speed increases by 9.8 m/s each second. 0 seconds = 0 m/s 1 second = 9.8 m/s 2 seconds = 19.6 m/s 3 seconds = 29.4 m/s 4 seconds = 39.2 m/s 5 seconds = 49.0 m/s

Changes in Direction Even if speed is constant, velocity can change (acceleration can occur) if there is a change in direction. Examples: Riding a carousel Biking around a curve Driving around a race track

Constant Acceleration Constant acceleration – a steady change in velocity Velocity changes by the same amount each second.

Calculating Acceleration For straight-line motion, calculate acceleration by dividing the change in velocity (the final velocity minus the initial velocity) by the total time. a = (v f - v i ) / t Note: Velocity can be treated as speed if the motion is in a straight line.

Graphing Acceleration Time is the independent variable, graphed on the x- axis (the horizontal axis). Speed is the dependent variable, graphed on the y-axis (the vertical axis). Slope= rise over run = change in y-axis over change in x-axis = change in speed over change in time = acceleration

Slope (of a speed-time graph) Since slope = acceleration… A positive slope indicates a positive acceleration (speeding up). A negative slope indicates a negative acceleration (slowing down). A horizontal line indicates an acceleration of zero. (The object is travelling at a constant speed.) A straight line indicates a constant acceleration.

Instantaneous Acceleration Instantaneous acceleration – how fast a velocity is changing at a specific instant