Scalar and Vector Quantities in Physics

• Scalar and vector quantities, distance, speed, velocity, and acceleration are fundamental concepts in physics
• The syllabus includes distance-time and speed-time graphs and equations of uniformly accelerated motion

Scalar and Vector Quantities

• Physical quantities are measurable and fall into two categories: scalar and vector quantities
• Scalar quantities are fully described by their magnitude e.g., mass of a body is 5.0 kg
• To express a scalar quantity completely, you need the unit and the numerical value
• Examples of scalars include mass, length, time, distance, density, volume, speed, temperature, energy, power, pressure, heat, charge, resistance, frequency, and angle
• Scalars can be added, subtracted, multiplied, and divided using simple arithmetic
• Scalar quantities are represented by their English letter e.g., mass is 'm', time is 't', speed is 'v'
• Vector quantities need magnitude and direction for complete clarity e.g., "displace a particle by 5 m to the east"
• To express a vector quantity, you need the magnitude, unit, and direction
• Examples of vectors include displacement, velocity, acceleration, momentum, force, torque, impulse, weight, electric field, and magnetic field
• The numerical value of a vector along with its unit gives the magnitude
• A negative sign indicates the reverse direction of a vector quantity
• Vector quantities follow specific rules for addition, subtraction, and multiplication
• Vectors are denoted by an arrow over the letter or using bold font e.g., velocity as or v, acceleration as or a, force by F or F

Rest and Motion

• Every object in the universe is in motion
• An object is at rest if it doesn't change position relative to its surroundings
• An object is in motion if it changes position relative to its surroundings
• A body can be considered a point particle if the distance it travels is much larger than its size
• One-dimensional motion occurs when a body moves along a straight line
• One-dimensional motion is also called rectilinear motion
• The example given is a train on a straight track or a stone falling vertically
• The position of a particle in one-dimensional motion can be represented on a straight line, with its coordinate changing over time
• The position of a pebble falling freely and vertically downwards at different times is provided as an example

Distance and Displacement

• Distance is the total length of the path traveled by a body
• Displacement is the length of a straight line from the initial to the final position, along with direction
• Distance depends on the path taken
• Distance is a scalar quantity, usually represented by the letter S
• The S.I. unit for distance is the meter (m), C.G.S. unit is the centimeter (cm)
• Displacement is the shortest distance from the initial to the final position, indicating direction
• Displacement is a vector quantity, represented by the symbol S
• The S.I. unit of displacement is the metre (m) and the C.G.S. unit is the centimetre (cm)
• Vectors have magnitude and direction, represented by an arrow in diagrams
• The arrow's tip indicates the direction and the length represents the magnitude using a specific scale
• 1. Magnitude of displacement is always less than or equal to the distance travelled
• Magnitude of displacement equals distance if motion is along a fixed direction
• Distance is the length of the path, so it is always positive
• Displacement depends on direction, so it can be positive or negative
• Displacement can be zero even if the distance isn't
• If a body returns to its starting point, displacement is zero, but the distance isn't

Speed

• Speed is the rate of change of distance with time
• Speed measures how fast a body is moving
• It is the distance traveled by the body in 1 second and it is a scalar quantity
• Represented by 'u' or 'v'
• Speed = Distance / Time (v = S/t)
• The unit of speed is the unit of distance divided by the unit of time
• S.I. unit for speed is meters per second (m/s)
• C.G.S. unit for speed is centimeters per second (cm/s)
• Uniform speed means covering equal distances in equal time intervals
• If a body moves with uniform speed v, the distance S traveled in time t is S = vt
• Changing speeds are called non-uniform or variable speed
• Instantaneous speed is the ratio of distance traveled in a short time interval to that time interval
• Speedometer measures instantaneous speed.
• Average speed is the ratio of the total distance traveled to the total time taken
• For a body moving with uniform speed, instantaneous and average speeds are the same

Velocity

• Velocity is the distance traveled in a specific direction
• Velocity is the rate of change of displacement with time
• Numerically equal to the displacement of the body in 1 s and a vector quantity and is represented by the symbol or v.
• Both magnitude and direction must be specified for velocity
• Two bodies have the same velocities if they move at the same speed and direction
• The unit of velocity is the same as speed, the S.I. unit is meters per second (m/s) and centimeter per second (cm/s).
• Uniform velocity means traveling equal distances in a particular direction in equal time intervals
• With uniform velocity, displacement s in time t is given by Displacement s = vt
• Variable velocity occurs if the magnitude or direction or both changes
• You have variable velocity if unequal distances are covered in a particular direction during equal intervals of time
• Instantaneous velocity is the velocity of a body at any instant
• The velocity of the body at any instant is called its instantaneous velocity
• Average velocity is the ratio of displacement to the total time taken when velocity changes over time in a particular direction
• Speed is a scalar, while velocity is a vector
• Speed tells how fast, velocity tells how fast and in what direction
• The magnetude of velocity is speed, but it can be positive or negative, but speed always is positive
• Average velocity can be zero even if average speed isn't
• A car moving in a circular track has a constent speed, but a changing direction

Acceleration and Retardation

• Acceleration is the rate of change of velocity with time
• Numerically, acceleration equals the change in velocity in 1 second
• Formula: Acceleration = Change in velocity / Time interval
• The S.I. unit of acceleration is meters per second squared (m/s²)
• The C.G.S. unit of acceleration is centimeters per second squared (cm/s²)
• A body moving in a straight line with initial velocity u changes to final velocity v in time t resulting in (v-u)/t
• Change in velocity is the difference between end and start
• Formula: v = u + at
• If v > u, acceleration 'a' is positive
• If v < u, acceleration 'a' is negative, called retardation
• Acceleration is a vector, direction is the direction of change in velocity
• Acceleration depends on its velocity
• Positive or negative sign of acceleration indicates the increase or decrease with time whereas the velocity determines direction.
• Uniform (or constant) acceleration has equal changes in velocity during equal intervals of time
• Free fall is an example
• Variable acceleration has unequal changes in velocity for the same time interval
• Value of g does not change with the mass of the object
• Objects of different masses dropped simultaneously will reach the ground because they have no friction and buoyancy due to air

Acceleration due to Gravity

• Acceleration due to gravity (g) is caused from free fall due to earth's gravitational pull
• Value of g is 9.8 m/s^2 or nearly 10m/s^2
• Velocity increases with time, so the acceleration is + g,
• Motion vertical upwards has a negative decrease with time
• The velocity of an object increased is 9.8m/s^2, therefore starting from rest, velocity after 1s is 9.8m/s