Kinematics
Lesson
8 half-hour periods
Relevant Chapter in Textbook
Chapter 21, Additional Mathematics, Panpac
Assignments
Two
Graded Assessment
Term 2, week 7, 1 hour, maximum score = 40, together with Integration
Objectives
By the end of this unit, you should be able to:
1. Apply differentiation and integration to kinematics problems involving
· Displacement / total distance travelled
· Velocity
· Acceleration of a particle moving in a straight line with variable or constant acceleration
2. Use and sketch the x-t and v-t graphs
1. Displacement, Velocity and Acceleration (Part I)
Objective
In this lesson (2 periods), you will learn about straight-line motion of a body using calculus knowledge.
Definition
1. Kinematics运动学
Kinematics is the study of the motion of a body without considering the cause of its motion.
We shall restrict our study to particles moving along a straight line (bi-directional motion双向移动 or one-dimensional motion 唯度移动) with a fixed point O on the line. One side of O is designated as positive and the other, negative.
2. Displacement位移
Usually denoted by s, the displacement of a particle is the position (positive or negative) of the particle with respect to a fixed point, usually denoted by O.
When s = 0, it means that ______
______
When s < 0, it means that ______
______
When s > 0, it means that ______
______
3. Velocity速度
With reference to the displacement-time graph,
Instantaneous velocity is .
Therefore, velocity v = or, .
We say that velocity, v, is the rate of change of displacement with respect to time, t.
When = 0, it means that ______
______
When < 0, it means that ______
______
When > 0, it means that ______
______
4. Acceleration加速度
With reference to the velocity-time graph,
Instantaneous acceleration is .
Therefore, acceleration a = or , or, .
We say that acceleration, a, is the rate of change of velocity with respect to time, t.
When = 0, it means that ______
______
When < 0, it means that ______
______
When > 0, it means that ______
______
Example
Let’s analyse the motion of a particle, P, that displaces from O in the manner described by the equation: s (in metres) = 3t – t 2 – 2, where t is the time in seconds.
1. Where is P initially? ______
______
2. In which direction is P travelling initially? ______
______
______
3. Is P slowing down or picking up velocity in its entire motion?
______
______
4. How many times does P reach O throughout its entire motion? When?
______
______
5. What is the velocity of P when it first reaches O?
______
______
6. How many times does P come to rest? When? Where?
______
______
7. Is it a permanent or instantaneous one? How can you tell?
______
______
8. What is the velocity of P when it returns to O? Where is it heading toward?
______
______
9. Describe the motion of P after it passes O the second time. Justify.
______
______
10. Sketch the path of P on the line provided. Label the values of t and v that you know.
Answers
1. When s = 0, it means that the particle is at the fixed point O. This does not imply that it is at rest, as that is determined by the value of .
2. When s < 0, it means that the particle is at the side of fixed point O which is defined as negative displacement. This does not imply anything about the direction in which it is travelling, as that is determined by the value of .
3. When s > 0, it means that the particle is at the side of fixed point O which is defined as positive displacement. This does not imply anything about the direction in which it is travelling, as that is determined by the value of .
4. When = 0, it means that the particle is at rest. It could be either an instantaneous state of rest or a permanent state of rest. It does not imply anything about the position of the particle, as that is determined by the value of s. If the particle is instantaneously at rest, the displacement could be a maximum or minimum, or it is the case of stationary point of inflexion.
5. When < 0, it means that either the particle is on the positive side of fixed point O and moving towards O, or, the particle is on the negative side of fixed point O and moving away from O.
6. When > 0, it means that either the particle is on the positive side of fixed point O and moving away from O, or, the particle is on the negative side of fixed point O and moving towards O.
7. When = 0, it means that either the particle is moving at a constant velocity (i.e. = c), or, it is at rest (i.e. = 0 too). It also means that velocity has a stationary value.
8. When < 0, it means that either the particle is slowing down as it travels in the direction designated as positive, or, it is picking up velocity as it travels in the direction designated as negative.
9. When > 0, it means that either the particle is picking up velocity as it travels in the direction designated as positive, or, it is slowing down as it travels in the direction designated as negative.
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