- A scalar quantity is one which has SIZE ONLY (e.g. distance). It is always positive (+ve).
- A vector quantity is one which has BOTH SIZE AND DIRECTION (e.g. momentum). It can either be positive or negative (-ve).
In the exam you need to be familiar with the terms 'displacement', 'velocity' and 'acceleration' for motion in a straight line
Also you need to be able to construct and interpret speed/distance/time graphs for a given journey or specified (e.g. the area under the graph represents distance travelled).
Displacement is a word used to represent the distance travelled from a fixed point. This displacement may be positive or negative to indicate movement to the left or right of the initial starting point.
The unit of measurement for displacement is metres (m) and is represented by the letter 's' in suvat equations.
A displacement/time graph will often be abbreviated to a (t,s) graph.
Velocity is a word used to define the speed of a body in a specific direction. For example to cars travelling in opposite directions of the motorway will have the same SPEED of 70mph, however one will have v = +70mph (north) and one will have v = -70mph (south).
The unit of measurement for velocity is metres per second (9.8ms-1) and is represented by the letter 'v' or the differential .
A velocity/time graph will often be abbreviated to a (t,v) graph.
Acceleration is a measure of the rate of change of velocity. As with displacement and velocity the acceleration of a body can be -ve or +ve to represent the direction in which acceleration is taking place.
A -ve acceleration is often called DECELERATION (or a retardation).
The unit of acceleration is metres per second per second (ms-2) and is represented by the letter 'a' or the differential .
An acceleration/time graph will often be abbreviated to a (t,a) graph.
Terminology: Average speed or average velocity
Average Speed or Average Velocity
The average speed or average velocity can be calculated for a journey which is made up of many parts. For example a train might make several stops at different railway stations along its route, however an average speed for the journey as a whole could be calculated.
Average speed =
Average velocity =
Motion due to gravity
Motion due to gravity
When a particle is released in mid air it naturally falls to the ground. Gravity causes the particle to accelerate at a constant rate of a = 9.8ms-2. This acceleration acts downwards as the particle is displaced downwards (or below) it’s original starting position.
It is convential to label things travelling downwards as having a –ve velocity, and as such its conventional to represent acceleration due to gravity as a = -9.8ms-2.
Units – the size of a force is measured in NEWTONS (N)
Body at rest – A body will remain in a state of rest OR motion with constant velocity in a straight line if the resultant of all forces acting upon it is zero.
Equilibrium – A body that remains stationary, or at constant velocity, when being acted upon by two or more forces is said to be in equilibrium.
Resistance – A resistive force, or frictional force will always act in the direction that opposes motion.
Net force – When several forces are acting in the same parallel direction, the net force or resultant force represents the difference between the sum of the forces in each opposite direction.
Newton's 3 laws of motion
Newton’s 3 laws of motion
1st. An object at rest will remain at rest unless acted on by an unbalanced force.
OR - An object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
2nd. When a resultant force of F Newtons acts upon a body of mass m kg, it causes it to accelerate in the direction of the resultant force at a constant rate according to the formula F = ma.
3rd.For every action there is an equal and opposite re-action.
As you are aware when an object is released from rest in mid air it falls to the ground in a vertical straight line. The object will accelerate at a constant rate of 9.8ms-2. You are also aware that if an object is accelerating then there must be some resultant force acting upon it, which in this case is the force of gravity. This is an invisible force which acts on all objects causing them to accelerate downwards at the same constant rate.
Weight – The weight of an object is an invisible force which represents the gravitational pull that is acting upon it. The simple formula to calculate the weight of an object is (where weight is in Newtons):
Weight = mass x gravity
(W = mg)
Forces that are acting at an angle need to be resolved into their horizontal and vertical components using basic trigonometry.
A force of F Newtons acting at an angle of q° to the upward horizontal can be resolved into a horizontal and vertical component, as shown below:
When a force is applied to a body, the frictional force that exists between the body and surface will become evident. This frictional force has the ability to increase or decrease according to the size of the force that is being applied. A small force will generally not cause the block to move. However there will be some limiting, or maximum or threshold, value of this frictional force, Fmax. To cause the body to move the applied force must exceed this limiting value.
There is a situation that commonly arises whereby the force being applied to the body is matched by this maximum, or limiting, value of the frictional force Fmax. At this point the body is on the point of moving and is said to be in a state of limiting equilibrium.
Forces applied at an angle – When the applied force is done so at an angle to the horizontal it complicates things in relation to calculating the size of the normal contact force. The upward (or downward) component of this applied force will have the effect of reducing (or increasing) the size of the contact force, thus affecting friction.
Smooth surface – Where the friction between the surface and the object is negligible and hence ignored.
Rough surface – Where the friction between the surface and the object is significant and hence needs to be considered.
Normal contact force (R) – A force which is applied to an object by the surface, perpendicularly.
We need to be able to consider objects that are travelling up or down a plane which is inclined at an angle to the horizontal. These problems offer a greater challenge to solve and so it is helpful in the exam if you draw a clear diagram.
- The weight of the block will always act vertically down
- The normal reaction, R, will always act at right angles to the plane.
- Friction will always oppose the direction of motion and as such can either run directly up or down the plane, depending on the direction of motion.
- In (almost) all cases the block will remain in contact with the plane and as such we will always start by resolving perpendicular to the plane.
For these problems we look at the 2 perpendicular directions which are parallel and perpendicular to the plane.
There is a unit of work which looks at how systems of connected particles react when released from rest.
-The strings that are used to attach the particles are referred to as LIGHT and INEXTENSIBLE. This means the weight of the string is negligible and can be ignored. It also means the strings do not stretch.
-The tension in any give string has the same value at both its ends. Tension always acts towards the centre of the string.
-Particles which are connected will accelerate at the same rate and will always travel with the same velocity, provided the ring remains taught.
-You can apply F = ma to either the system OR the individual particles.
Linear momentum is a quantity which is associated with the motion of an object in a straight line. The momentum of an object is found using the formula:
Momentum = mass x velocity
(Momentum = mv)
The units for momentum are Ns (Newton seconds) and as the formula involves velocity it means direction is important. In other words momentum will be a vector quantity.
A very useful application of the property of momentum is to study how 2 objects behave when they are involved in a head-on collision. The rule of conservation of momentum states that MOMENTUM BEFORE = MOMENTUM AFTER.
For any body that is involved in such a collision, CHANGE IN MOMENTUM = MOMENTUM AFTER – MOMENTUM BEFORE.
In many mathematical problems, acceleration is not a constant value. The velocity and acceleration for the majority of bodies are constantly changing with time. We can apply calculus to model situations where this is the case.
Rule 1: The displacement of a body, x metres at time t seconds can be given in the form x = f(t).
Rule 2: The velocity is defined as the rate of change of displacement with respect to time. In other words v =
Rule 3: The acceleration is defined as the rate of change of velocity with respect to time. In other words a = (or a = d2x/dt2).
Non-uniform acceleration: The reverse problem
Non-uniform acceleration: The reverse problem
In a similar question we may have information regarding the acceleration of a particle at time t seconds and may wish to find an expression for its velocity and its displacement at time t seconds.
In these cases it seems a natural extension to apply integration to work backwards through the problem.
Rule 4: An expression for the velocity of a body can be found by integrating the expression for its acceleration, and hence .
Rule 5: An expression for the displacement of a body can be found by integrating the expression for its velocity and hence .
Rule 6: The area under a velocity/time graph represents the distance travelled. Applying our knowledge of calculus means the displacement of an object between times t1 and t2 is given by the definite integral .