Vocabulary average speed constant speed coordinates coordinate system displacement instantaneous speed instantaneous velocity origin position rate slope time vector velocity

Space and position In physics, the word position refers to the location of an object at one instant. A position is always specified relative to an origin. The net change in position relative to the origin is called displacement.

Position and distance Distance is related to, but different from, position. Distance is a measure of length without regard to direction.

Position in three dimensions Space is three dimensional, so position must also be a three- dimensional variable. Any position in space can be precisely specified with three numbers called coordinates.

Positive and negative Allowing x, y, and z to have positive and negative values allows coordinates to locate any position in all of space.

One dimensional problems In three-dimensional space, position is a vector. A vector is a variable that contains all three coordinate values. Motion in a straight line is easiest to analyze because it is one dimensional. However, even in one dimension there is an origin and positive and negative values are possible.

Speed and distance Speed is the rate at which distance changes. In physics, the word rate means the ratio of how much something changes divided by how long the change takes. Constant speed means the same change in distance is traveled every second.

Calculating speed The change in position is a distance traveled in a given amount of time. To calculate the speed of an object, you need to know two things: the distance traveled by the object the time it took to travel the distance

Calculating speed Since speed is a ratio of distance over time, the units for speed are a ratio of distance units over time units.

Calculating speed in meters per second A bird is observed to fly 50 meters in 7.5 seconds. Calculate the speed of the bird in m/sec. You are asked for speed in m/s. You are given distance = 50 m; time = 7.5 s Use v = d ÷ t Plug in values and solve. v = 50 m ÷ 7.5 s ≈ 6.67 m/s

The velocity vector The velocity of an object tells you both its speed and its direction of motion. A velocity can be positive or negative. The positive or negative sign for velocity is based on the calculation of a change in position. Two cars going opposite directions have the same speed, but their velocities are different— one is positive and the other is negative.

The velocity vector Velocity is the change in position divided by the change in time.

Velocity Equations Velocity (v) is calculated by dividing the change in position (Δx) by the change in time (Δt).

Working with Equations An equation is a much more powerful form of model than a graph. While graphs are limited to two variables, equations can have many variables and can be used over a wide range of values.

Working with Equations Equations can also be rearranged to show how any one variable depends on all the others.

Calculating time from speed and distance How far do you go if you drive for 2 h at a speed of 100 km/h? You are asked for distance. You are given time in h and speed in km/h. Use d = vt. Solve. d = 2 h × 100 km/h = 200 km

Solving an equation To “solve” means to get a desired variable by itself on one side of an equals sign. Whatever you do to the left of the equals sign you must do exactly the same to the right. Get in the habit of solving an equation before you plug in numbers. More complex problems require you to substitute whole equations for single variables.

Solving an equation To solve this equation for distance (d): Multiply both sides of the equation by “t”. Multiplying by “t”on both sides of the equation allows you to cancel a t from the numerator and the denominator on the right side of the equation.

Position vs. time equation The equation says your position, x, is equal to the position you started at, x0, plus the additional amount you traveled, vt.

Calculating time from speed and distance A car moving in a straight line at constant velocity starts at a position of 10 meters and finishes at 30 meters in five seconds. What is the velocity of the car? You are asked for velocity. You are given that the motion is at constant velocity, two positions, and the time. Use x = x0 + vt, solve for v. x – x0 = vt x – x0 = v t Substitute numbers for variables: v = 30 m – 10 m = 4 m/s 5 s