6. VECTORS AND SCALARS

2.0 Scalar and Vector Quantities

Scalar Quantities:  Scalar quantities are those quantities which has only magnitude but no direction. Examples: Mass, length, density, volume, energy, temperature, electric charge, current, electric potential etc.

Vector Quantities: Vector quantities are those quantities which has both magnitude as well as direction.

Examples: Displacement, velocity, acceleration, force, electric intensity, magnetic intensity etc.

Vector Addition

While scalars can be added or subtracted easily (e.g. 30cm+ 50cm = 90cm and  3m-2m = lm), vector addition takes into consideration not only the magnitude but the direction of the vectors. The method of calculation also depends on whether the vectors are acting in a straight line or at an angle to each other, illustrated by the following typical cases. In addition, vectors can be added using; triangle law and parallelogram law. These vectors can be resolved into horizontal and vertical components.

a. i Two parallel forces acting in same direction

The result force R = 70+90 = 160N

ii. Two forces acting in opposite direction in a straight line

the resultant force R = 90N-70N = 20N

a.       Two forces acting perpendicularly (At right Angle or 900) to each other

The resultant R is the diagonal of the rectangle (If F1 and F2) are not equal to each other or the square of (If F1 = F2)

The resultant R is found using Pythagoras  theorem;

R2 =  or R =

2.1 DISPLACEMENT, VELOCITY AND ACCELERATION

Speed

Speed is defined as distance moved by an object per unit time. It is a scalar quantity and is measured in metre per second (m/s or ms-1).

Velocity

Velocity is defined as distance travelled in a specified direction per unit time.

OR

Velocity is defined as displacement over time. It is a vector quantity and is also measured in metre per second (m/s or ms-1).

Both speed and velocity have the same unit, formula and dimension.

v = s/t

where v = velocity(speed) in m/s

s = displacement (distance) in m

t = time in seconds.

Examples on speed and velocity

1.       An air force jet flying with a speed of 335m/s went past an anti-aircraft gun. How far is the aircraft 5s later when the gun was fired.

2.       A student walks a distance of 3km in 20 minutes. Calculate his average speed.

3.       A man cycles non-stop through a distance of 1.0km in 5minutes. Calculate his average speed.

4.       4. A car travels with a constant velocity of 45 km/hr for 20s. What distance does it cover in this time?

5.       A marble rolls 5.2 m in 1.8 s. What was the marble's average speed

 

Rectilinear Acceleration

The term rectilinear acceleration means the rate of increase of velocity along a straight-line path in a unit time. When the velocity of an object changes it could be said to accelerate or decelerate. Acceleration is defined as the increasing rate of change of velocity with time. Deceleration on the other hand is defined as the decreasing rate of change of velocity with time.

Deceleration is also called retardation or negative acceleration.

Acceleration (Deceleration) = Change in velocity

Time taken for change 

= F inal velocity - Initial velocity

        Final time - Initial time

Equations of Uniformly Accelerated Motion

Equations of motion for a body traveling along a straight line with uniform acceleration are as follows:

v = u + at --------------------(1)

s = ut + ½ at2 --------------------(2)

v2 = u2 + 2as ---------------(3)

Always Remember:

1. When an object moves or accelerates from rest, its initial velocity, u = 0.

2. When a body comes to rest or stops, its final velocity, v = 0.

3. When a body’s velocity is constant or not changing, its acceleration, a = 0.

Examples:

1.       A cyclist is travelling at 15 ms−1 at a distance of 18m, brakes so that she doesn’t collide with the wall. Calculate the magnitude of her deceleration.

2.       Leaving a bus stop, a bus reaches a velocity of 8.0ms−1 after 10s. Calculate the acceleration of the bus.

3.       A sprinter starting from rest has an acceleration of 5.0ms−2 during the first 2.0s of a race. Calculate her velocity after 2.0s.

4.       A train slows down from 60ms−1 to 20 ms−1 in 50s. Calculate the magnitude of the deceleration of the train.

5.       If a car starts from rest and moves with a uniform acceleration of 12m/s2 for 8s. What is the distance it's covers in the last 3sec of the motion?

6.       A car moving with a speed of 90km/hr was brought to rest by the application of brakes in 10s. How far did the car travel after the brakes were applied?

Exercise

1.       A body uniformly accelerates from rest at 8m/s2. In how much time will the body travel a distance of 2.5km?

2.       A particle starts from rest and moves with a uniform acceleration of 4m/s2. What is its velocity after covering a distance of 8m?

3.       A particle accelerates uniformly from rest at 6.0m/s2 for 8s and then decelerates uniformly to rest in the next 5s. Determine the magnitude of the deceleration.

4.       A body accelerates uniformly from rest at the rate of 3ms-2 for 8s. Calculate the distance covered by the body during acceleration.

5.       A bus driver moving at a velocity of 100km/hr suddenly sees a Goat Crossing the Highway 49m ahead. He hit hand on his brakes to get a maximum retardation of 8.0ms-2

a.       How far does he go before stopping?

b.       Can he avoid hitting the Goat?

6.       A rocket lifts off from rest with an acceleration of 20ms−2. Calculate its velocity after 50s.

7.       A car is travelling along a straight road at 8.0 ms−1. It accelerates at 1.0 ms−2 for a distance of 18 m. How fast is it then travelling?

8.       A train travelling at 20ms−1 accelerates at 0.50ms−2 for 30s. Calculate the distance travelled by the train in this time.