Rectilinear motion with variable acceleration

The Analysis of Rectilinear Motion

If we pay close attention to the motions in a bicycle racing, we many notices that there are two kinds of motion occurred,i.e.: rotational and translational motions. The examples of the rotational motions are the motion of the racer's leg pedaling the pedal, the motion of the bicycle's wheels and the motion of the pedaled pedal. The examples of the translational motion are the motion of the bicycle and the racer when they are moving away from the start line.Base on the illustration of bicycle racing above, we have some questions related to physics. How about the velocity and accelerations of the bicycle? How about the motion of the bicycle's wheel? For answering the questions correctly, you need to study the following discussion.

During the motion, the position of a body always changes. The moving body always has velocity and probably acceleration as well. How about the relationship of the displacement, velocity, and acceleration of body?

1. Position of a  body in a Plane or in space

by using a coordinate system we can determine the position of a body. there are several types of coordinate system; some of them are Cartesian and polar coordinate systems. In discussing the rectilinear motion, the Cartesian coordinate is preferred, both the two or three dimensional systems. On the other hand, the polar coordinate system is preferred to be used in discussing the circular motion.

In a coordinate system, the position of a body is represented by a position vector. Before discussing position vector further, we need to discuss the unit vector.

a. Unit Vector

A unit vector is a vector, which magnitude is one. It has no unit and its direction is along the axes of the coordinate. For the Cartesian coordinate system, the directions of the unit vectors are along the x, y, and z-axes.

In a Cartesian coordinate system, unit vectors are usually denoted as i, j and k for the unit vectors along the x, y, and z-axes respectively, are:

Ax = A Cos α and Ay = A Sin α

In the form of unit vectors, vector A could be written as follows:

A= A_x i + A_yj

If vector A is spatial vector then

A= A_x i + A_yj +A_zk

b. Position Vector

A position vector is a vector that represents the position of a particle in a plane or in space.

The position of a  body in a  planar surface could be represented by a  position vector of r,  that is a  vector drawn from the origin to the position of the body.  Look at Figure 1.5.  The  position  vector of  point P(x, y) could  be  written as fo1lows: 

r = xi +yj

Whereas,  a  position vector in  a  three-dimensional space  could  be written as follows:

r = xi +yj +zk

In a rectilinear motion,  displacement is defined as the change of position of a body.  For example,  at time t₁,   a body is at point P(x, y);  this position is represented by position vector r₁. A few seconds later, at time r₂, the body is at point Q (x_1,y₂), which position is represented by position vector r₂. A vector. A vector (say Δr), which is drawn from P to Q can be written as follows.
Δr =  r₂-  r₁
     = (x₂i - y₂j) - (x₁i - y₁j)
     = (x₂ - x₁)i  + (y₂ - y₁)j
     = Δxi + Δyj

Example

A body is initially at  r₁ = 3i - 2j . then it moves to  r₂ = -i + 4j. Determine the magnitude of displacement of the body.

Answer :
The initial position of the body is  r₁ = 3i - 2j; the final position of the body is  r₂ = i - 4j. The displacement of the body is

 Δr =  r₂-  r_{1 
       =} (-1 - 3) i + 4 -(-2) j = -4i -2j

Exercise 1.1

1. A body move from point P (1, 0, 1) to point Q (5, 4, 3). Determine the displacement vector of the body and its magnitude.
2. A rabbit moves according to the equation x = t² and y = 5t. Determine: (a) the components of velocity at t = 0 second and at t = 4 seconds, (b) the average velocity vector, and (c) the magnitude and the direction of the average velocity.
3. The position of a moving particle is represented by r = 4t -3t² + t³, where r is n meter and t is in second. Determine the average velocity from t₁ = 0 to t₂ = 2 seconds.
4. Position vector of is represented by r = {(cos 2t) i + (4 sin2t) j)} m. Determine the particle velocity at t = 2 seconds.
5. The formula of displacement of a body, which is moving in a straight line, is x = -t² + 4t – 20, where r is in meter an t is in second. Determine the initial velocity and the acceleration of the body.
6. A particle is moving in a certain direction in such a way that its displacement components are given by x = 4t² + 2 and y = 4t + 2t², respectively. Determine the magnitude of the particle's acceleration at t = 2 seconds.
7. A ball is moving with the initial velocity of 20 m/s. If the acceleration of the ball is a = (4t + 2) m/s2, what are the equations of velocity and position of the ball? (its initial position is (0, 0)).

B. Analysis of circular Motion

A circular motion is a motion path is in the form of a circle. For examples, the motions of bicycle wheels, phonograph record, and helicopter's propeller. In the circular motion, the path's radius r is always constant, so that the component of the polar coordinate that determines the position of a circularly moving point is the traveling angle, which is Ө. A bicycle wheel rotating with respect to the fixed axes through point O. The line OP is a fixed line on the wheel that rotates with the same speed as the wheel. The position of point P is determined by the angle Ө formed by the line OP and the positive x-axis.

If the angle Ө is measured from the positive x-axis counter clockwise, then the Ө is negative. By definition, the angle Ө (in radian) is calculated using the formula:

Ө = s/r  rad

Ө = s/r = 2πr/r = 2π rad

However, one complete rotation is equal to 360^o. Then,

1 rotation = 360^o =  2π rad

1 rad = 360^o /2π = 57.3^o or 1^o = 0.01745 rad

The velocity possessed by a body undergoing circular motion is called angular velocity. Angular velocity is defined by the rotation angle covered in a unit interval of time. The unit of angular velocity is radian per second (rad/s).

Example:

A phonograph record with the diameter of 20 cm rotates with the angular speed of 30 revolutions per minute. Calculate the magnitude of the angular velocity and linear velocity of a point situated on the perimeter of the phonograph record.

Answer:

The magnitude of the angular velocity of the record:

ω = 30 rpm = 30 rotations per minute = 30 x 2π rad/ 60 s

     = π rad/ s

The magnitude of the linear velocity v = ω r = ½ ωd

v = π [ ½ (π rad/s) ( 20 cm)]

   = 31.4 cm/s

You Need to Know

Roller coaster is a high-speed train that passes through rectilinear and circular paths. When the train passes circular path it will undergo centripetal acceleration toward the center of the circle. To keep the train stay on track, it needs a big centripetal force obtained by accelerating the train. Thus, the passengers of roller coaster have to be safety strapped to the chair, to avoid them from being thrown away due to the high speed.

Case study

Between Soccer and Physics

What the soccer players do is very closely related to physics. For example, when he kicks the ball to the goal post, he should predict correctly the velocity as well as the elevation angle of the ball. If the elevation angle and the velocity are too great, the ball will miss the goal post. On the contrary, if the elevation angle and the velocity are too small or low, the ball will not reach the goal post.

Now is your turn to analyze and explain the following problems by using physical concepts.

1. Why does the path of the ball from a parabolic pattern?
2. Why is it difficult for the goalkeeper to block penalty kick?

physics rectilinear motion rigid body vector

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