Distance, displacement, speed, velocity, acceleration, motion graphs, Newton's three laws, linear momentum, and conservation of momentum in collisions.
These four quantities form two scalar-vector pairs. Distinguishing them precisely is important in physics.
| Scalar | Vector | Definition |
|---|---|---|
| Distance | Displacement | Distance is the total path length travelled. Displacement is the straight-line distance from start to finish in a specified direction. |
| Speed | Velocity | Speed is the distance travelled per unit time. Velocity is the displacement per unit time (or rate of change of displacement); it has direction. |
Both speed and velocity use units of m s⁻¹.
Acceleration is the rate of change of velocity, or the change in velocity per unit time:
where is initial velocity and is final velocity. Units: m s⁻².
The gradient of a displacement-time graph equals velocity.
The gradient of a velocity-time graph equals acceleration. The area under a velocity-time graph equals displacement.
To find displacement from a v-t graph, calculate the area of each region geometrically (rectangle for constant velocity, triangle for acceleration/deceleration). Areas below the time axis represent motion in the opposite direction and count as negative displacement.
Aristotle (4th century BC) argued that a continuous force is needed to keep an object moving: the faster the force applied, the faster the object moves (). He believed that objects naturally come to rest and that rest is the natural state of all things.
This view was eventually shown to be wrong. The reason everyday objects slow down is friction and air resistance, not a fundamental tendency to stop. Newton later showed that a force changes velocity, not velocity itself.
An object remains at rest or continues moving at constant velocity unless acted on by a resultant external force. This is the law of inertia, objects resist changes to their state of motion.
Inertia is the tendency of an object to resist any change in its state of rest or uniform motion. A more massive object has greater inertia; it requires a larger force to produce the same acceleration.
The practical consequence: if something is accelerating, there must be a non-zero resultant force. If it moves at constant velocity, the forces are balanced.
The resultant force on an object equals its mass multiplied by its acceleration:
where is in newtons, in kilograms, and in m s⁻². The acceleration is in the same direction as the resultant force.
Every force has an equal and opposite reaction force, acting on a different object. The two forces in an action-reaction pair are equal in magnitude, opposite in direction, and act on different bodies, they cannot cancel each other.
Common examples of Newton's Third Law in dynamic systems:
A 70 kg test dummy is in a car travelling at 26 m s⁻¹. The car crashes and stops in 0.1 s.
Part (i), Initial momentum:
Part (ii), Force from seatbelt:
The dummy decelerates from 26 m s⁻¹ to 0 in 0.1 s.
The seatbelt exerts a force of 18,200 N on the dummy.
Momentum is the product of mass and velocity:
Units: kg m s⁻¹ (equivalent to N s).
Momentum is a vector, it has the same direction as the velocity.
Force equals the rate of change of momentum:
This form is more general than because it works even when mass changes.
In a closed system (no external forces), the total momentum before a collision equals the total momentum after:
This holds for all collisions, elastic and inelastic, as long as no external forces act.
An 8 kg ball moving east at 10 m s⁻¹ collides with a 2 kg ball moving west at 5 m s⁻¹. After the collision they stick together. Find their common velocity.
Take east as positive.
Total momentum before:
After collision (combined mass = 10 kg):
Assign a positive direction at the start of every momentum problem and stick to it. A velocity in the opposite direction gets a negative sign. If your final velocity comes out negative, the object is moving in the direction you called negative.