Vectors, motion and inclined planes

Choose a positive direction

Suppose a trolley moves east, slows down, then rolls west. A watch and a metre scale give you everything you need to describe this motion, provided you say which way counts as positive. Choose east. Position is the coordinate you read from the scale. Displacement (spostamento) is final position minus initial position. A trolley can travel a considerable distance and still have zero displacement if it ends where it started. Distance adds the lengths of all the pieces of its journey.

Velocity tells you how quickly position changes and in which direction. Average velocity is displacement divided by elapsed time. Speed is the magnitude of velocity at an instant, and average speed is total distance divided by elapsed time. To see why the distinction matters, imagine walking three metres east and one metre west in four seconds. Your average velocity is half a metre per second east. Your average speed is one metre per second, since your feet have covered four metres.

Read the axes before the line

On a position–time graph, choose two points on a straight segment and divide the change in position by the change in time. The slope gives velocity. A horizontal segment describes a stationary object, even if it lies far above the horizontal axis. A straight segment sloping down describes constant negative velocity. On a curve, a tangent gives instantaneous velocity. Two points separated by a long time give only an average over that interval.

On a velocity–time graph a horizontal segment describes constant velocity. The slope now gives acceleration (accelerazione), while the signed area between the line and the time axis gives displacement. Split straight-edged regions into rectangles and triangles. Areas below the axis count negatively towards displacement and positively towards distance. If a line crosses the axis, split it there before adding distances. The object can have zero velocity at the crossing while the sloping line still gives a non-zero acceleration.

Draw the forces acting on one body

Newton's second law connects the forces on a body with its acceleration. Draw the body alone and label each external interaction. Earth pulls it down through its weight, with magnitude $mg$. A smooth surface pushes perpendicular to itself. A taut string pulls along its own length. A rough surface can exert a frictional force along the surface. Kinetic friction opposes sliding relative to the surface. The reaction force that the body exerts on a support acts on the support, so leave it off the diagram of the body.

Choose axes to match the allowed motion. For a block on an inclined plane (piano inclinato), take one axis down the slope and the other perpendicular to it. If the slope makes an angle $\theta$ with the horizontal, the downhill component of weight is $mg\sin\theta$. The component pressing into the plane is $mg\cos\theta$. Picture gradually lowering the ramp. On the level the downhill component vanishes, and the perpendicular component equals the whole weight. This thought experiment recovers the correct choice of sine and cosine.

Turn the force diagram into a prediction

Add the signed components along the slope. For a sliding block moving downhill with a string pulling uphill, tension and kinetic friction both enter with a minus sign when downhill is positive. Divide the net force by mass to find acceleration. The perpendicular acceleration is zero while the block follows a straight ramp. The normal reaction then balances the forces pressing into the surface. A string at an angle to the ramp would affect this balance as well as the downhill sum.

Check the signs against the motion before stating the result. A negative acceleration means the velocity changes towards the negative direction each second. A block travelling uphill with downhill acceleration slows during its climb and may reverse later. At an instant of rest the forces still determine what happens next. If friction is present, decide whether static friction can hold the block before applying a sliding-friction model. Finish with units. Dividing a force in newtons by kilograms must give metres per second squared.