Instruction

The phenomenon. A crossing guard steps into a school-zone crosswalk. Near a California school, when children are present, the default speed limit is 25 mph (California Vehicle Code section 22352). Whether a driver can stop in time depends on four quantities that physics defines precisely: position, displacement, velocity and acceleration. This unit builds them into a model you will use to design a safer crossing.

Units first. Physics works in SI units. One mile is exactly 1,609.344 m, so 1 mph = 0.44704 m/s, and 25 mph = 25 × 0.44704 = 11.18 m/s. A car at the school-zone limit covers about 11 m every second, the length of two and a half cars.

Position and displacement. Position x is where an object is, measured from a chosen origin along an axis with a chosen positive direction. Displacement is the change in position, Δx = x_f − x_i. It has a sign, so it is a vector in one dimension. Distance is the total path length and is never negative. Example: you walk 600 m east to school in 8 min, then 200 m back west to a friend’s house in 2 min. Distance = 800 m, but displacement = +400 m (east).

Velocity and speed. Average velocity is displacement divided by time, v_avg = Δx/Δt; average speed is distance divided by time. For the walk, average speed = 800 m / 600 s = 1.33 m/s, but average velocity = 400 m / 600 s = 0.67 m/s east. On a position-time graph, velocity is the slope. A steeper line means faster motion, a horizontal line means the object is at rest, and a downward slope means motion in the negative direction.

Acceleration. Acceleration is the rate of change of velocity, a = Δv/Δt, in m/s per second (m/s²). On a velocity-time graph, acceleration is the slope.

Worked example: leaving the crosswalk. A car waits at the line, then speeds up steadily to 12 m/s in 4 s, cruises at 12 m/s for 6 s, and brakes steadily to rest in 4 s at the next corner.

• Slopes give accelerations: (12 − 0)/4 = +3.0 m/s², then 0, then (0 − 12)/4 = −3.0 m/s².
• The area between the graph and the time axis is the displacement, because each thin strip is velocity × time. Triangle: ½(4)(12) = 24 m. Rectangle: 6 × 12 = 72 m. Triangle: 24 m. Total: 120 m, and the average velocity is 120/14 = 8.57 m/s.

The position-time graph of the same trip is below. While the car speeds up, the slope grows, so the curve bends upward. While it cruises, the graph is a straight line of slope 12 m/s. While it brakes, the slope shrinks to zero and the curve levels off at 120 m.

Common misconceptions. (1) A motion graph is not a picture of the road. A rising line on a velocity-time graph does not mean the car is going uphill. (2) Negative acceleration does not always mean slowing down. An object slows down when velocity and acceleration have opposite signs. A car moving in the negative direction and speeding up has negative velocity and negative acceleration. (3) Zero velocity does not mean zero acceleration. A ball thrown straight up has v = 0 at the top but is still accelerating downward at 9.8 m/s².

Driving question for the unit. A driver at 25 mph sees the crossing guard. How far does the car travel before it stops, and what should a yellow light’s duration, or a school zone’s speed limit, be so that stopping is possible? Keep your first guess. You will test it with a model by the end of the unit.

Formative check

Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.