Translational Motion

One dimension means the magnitude of length or distance only. Two dimensions means the length or distance on a 2D plane (xy coordinates) while three dimensions signifies length or distance in 3D space (xyz coordinates). Four dimensions is length or distance in 3D space at a given time (xyzt coordinates).Vectors are numbers that have magnitude and direction. Vector quantities include displacement, velocity, acceleration, and force. Scalars are numbers that have magnitude only and no direction. Scalar quantities include distance, speed, energy, pressure, and mass. The difference between a vector and scalar quantity can be quite pronounced when there is a nonlinear path involved. For example, in the course of a year, the Earth travels a distance of roughly 940 million kilometers. However, because this is a circular path, the displacement of the Earth in one year is zero kilometers. Vector components refer to the portion of the vector in a given direction.

SOH: sinθ = opposite / hypotenus.

CAH: cosθ = adjacent / hypotenus.

TOA: tanθ = opposite / adjacent.

sohcahtoa

axis-vector-components

vector-vector-components

gravity-components-on-slope

You can only directly add vectors if they are in the same direction. To add vectors in different directions, you must add their x, y and z components. The resulting components make up the added vector. The vector sum of all components of a vector equal to the vector itself. Operations involving a vector and a vector may or may not result in a vector, as the kinetic energy from the square of vector velocity results in scalar energy. Operation involving a vector and a scalar always results in a vector while operation involving a scalar and a scalar always results in a scalar. The sum or difference of two or more vectors is called the resultant of the vectors. One way to find the sum or resultant of two vectors A and B is to place the tail of B at the tip of A without changing either the length or the direction of either arrow. In this tip-to-tail method, the lengths of the arrows must be proportional to the magnitudes of the vectors. The vector sum A + B is the vector joining the tail of A to the tip of B and pointing toward the tip of B.

Head-to-tail_Addition

When adding vectors, always add tip-to-tail. Another method for finding the resultant of several vectors involves breaking each vector into perpendicular components. In most cases, these components are horizontal and vertical (x– and y-components, respectively). However, in some instances, such as inclined planes, it may make more sense to define the components as parallel and perpendicular (|| and ⊥, respectively) to some other surface. Given any vector V, we can find the x– and y-components (X and Y) by drawing a right triangle with V as the hypotenuse. If θ is the angle between V and the x-component, then

X = Vcosθ

Y = Vsinθ

Subtracting one vector from another can be accomplished by adding a vector with equal magnitude, but opposite direction to the first vector. This can be expressed mathematically as

A – B = A + (–B),

where –B represents a vector with the same magnitude as B, but pointing in the opposite direction. Vector subtraction may also be performed on the component vectors first and then combined to create a final vector. As with vector addition, the x-component of the resultant vector is the difference of the x-components of the vectors being subtracted. Similarly, the y-component of the resultant vector is the difference of the y-components of the vectors being subtracted. Notice that when you subtract vectors, you are simply flipping the direction of the vector being subtracted and then following the same rules as normal by adding tip-to-tail.

Speed is a scalar quantity as it has no direction and is the rate of change in distance. Velocity, on the other hand, is a vector quantity with direction and is the rate of change in displacement.

Speedaverage = uavg = 1/t

Velocityavg = Vavg = s/t

Instantaneous speed is the speed at an instant (infinitesimal time interval) while instantaneous velocity is the velocity at an instant (infinitesimal time interval). Instantaneous speed equals instantaneous velocity in magnitude. Instantaneous velocity has a direction but instantaneous speed does not. The direction of instantaneous velocity is tangent to the path at that point. Acceleration is the rate of change in velocity.

aavg = (Vf – Vi)/t

It is uniformly accelerated motion along a straight line. If acceleration is constant and there is no change in direction, then the value of speed/velocity, distance/displacement are interchangeable.

s =vavg at

vavg = (Vf – Vi)/2

Vf2 = Vi2 + 2as

s = ½ at2 + vit

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