RI Chap 3 Motion and Forces Lecture Notes

3 MOTION & FORCES H2 Physics 9478 Content Page 3.1 Kinematics 2 3.2 Uniformly accelerated linear motion 5 3.3 Mass and linear momentum 11 3.4 Newton’s Laws of motion 12 Learning Outcomes Candidates should be able to: (a) show an understanding of and use the terms position, distance, displacement, speed, velocity, and acceleration. (b) use graphical methods to represent distance, displacement, speed, velocity, and acceleration. (c) identify and use the physical quantities from the gradients of position- time or displacement-time graphs, and areas under and gradients of velocity -time graphs, including cases of non- uniform acceleration. (d) derive, from the definitions of velocity and acceleration, equations which represent uniformly accelerated motion in a straight line. (e) solve problems using equations which represent uniformly accelerated motion in a straight line, e.g. for bodies falling vertically without air resistance in a uniform gravitational field. (f) show an understanding that mass is the property of a body which resists change in motion (inertia). (g) define and use linear momentum as the product of mass and velocity. (h) state and apply each of Newton’s laws of motion: 1st law: a body at rest will stay at rest, and a body in motion will continue to move at constant velocity, unless acted on by a resultant external force; 2nd law: the rate of change of momentum of a body is (directly) proportional to the resultant force acting on the body and is in the same direction as the resultant force; and 3rd law: the force exerted by one body on a second body is equal in magnitude and opposite in direction to the force simultaneously exerted by the second body on the first body. (i) recall the relationship resultant force 𝐹𝐹 = 𝑚𝑚𝑚𝑚 for a body of constant mass, and use this to solve problems.

Pag e | 2 3.1 Kinematics  Introduction The study of the motion of objects, with the associated concepts of force and energy, is called mechanics. Mechanics can be further divided into two parts: 1. kinematics which describe how objects move and 2. dynamics which deal with force and why objects move as they do. Motion can be categorized into three types: • translational, • rotational and • vibrational. In this chapter, we are concerned only with translational motion and will treat the moving object as a particle, regardless of its size. Strictly speaking, a particle is a point -like object with mass but no size. However, we can still apply the particle model to objects such as a ball or a car, provided the positions of the objects refer to their centres of mass. We begin our study with rectilinear motion , which is motion in one dimension or motion in a straight line, and then proceed to projectile motion, which is an example of motion in a two-dimensional plane. Circular motion is another example of motion in a plane and that will be covered in a later chapter. All measurements of distance or speed are made relative to a frame of reference. Unless otherwise specified, the frame of reference is assumed to be that of a stationary observer on the Earth. Once the frame of reference is specified, it is then represented by a coordinate system. For motion in a plane, a pair of mutually perpendicular axes is chosen. The most common pair of axes is the horizontal and vertical axes, usually labelled as the x- and y- axes respectively. The point of launch or start of motion is usually designated as the origin and the directions for the x- and y- axes are chosen arbitrarily from the origin.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 3  Defining the Basics Scalar Vector Distance, x Displacement, s The total length of path an object travels. SI unit is metre (m). The distance moved in a specified direction from a reference point. SI unit is metre (m). Speed, v Velocity, v The instantaneous speed of an object is defined as the rate of change of distance travelled with respect to time. dx dt v = SI unit is metre per second (m s−1). The instantaneous velocity of an object is defined as the rate of change of displacement with respect to time. ds dt v = SI unit is metre per second (m s−1). Average speed refers to the total distance travelled over total time taken. x t v ∆ ∆ = Average velocity refers to the change in displacement over total time taken. s t v ∆ ∆ = Acceleration, a No scalar equivalent of acceleration. The instantaneous acceleration of an object is defined as the rate of change of velocity with respect to time. dva dt= SI unit is metre per second squared (m s–2). Average acceleration refers to the change in velocity over time taken. va t ∆= ∆

Pag e | 4  Graphs in Kinematics Graphs are very useful in representing the changes that occur during the motion of an object. There are three possible graphs that can provide useful information: • displacement-time (s-t) graph • velocity-time (v-t) graph • acceleration-time (a-t) graph Information from the graphs is often obtained from 1. direct reading of a point on the line / curve, 2. the gradient of the graph, and 3. the area under the graph. The physical quantities obtained depend on what is being plotted on the graph. Always look at the axes of a graph very carefully. s-t graph v-t graph a-t graph Gradient at a point instantaneous velocity instantaneous acceleration no physical significance Area under graph no physical significance = change in displacement = change in velocity ds dt dv dt = ∆∫v dt s = ∆∫a dt v

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 5  Conversion between graphs In order to analyze the motion of objects fully, it is important to be able to convert the graphs from one form to another. Case 1: Constant velocity Fig. 3.1(a) Fig. 3.1(b) Fig. 3.1(c) Case 2: Increasing velocity under constant acceleration Fig. 3.2(a) Fig. 3.2(b) Fig. 3.2(c) Graphs for an object moving with constant velocity. Graphs for an object moving with constant acceleration. 3.2 Uniformly accelerated linear motion  Kinematics equations The motion of an object whose velocity is increasing at a steady rate is called uniformly accelerated motion. The graph below shows the velocity-time graph of an object moving with constant acceleration. Its initial velocity is u and its velocity at time t later is v. Fig. 3.3 s t 0 v t 0 a t 0 s t 0 v t 0 a t 0

Pag e | 6 Acceleration a = gradient of v-t graph vua t −= Hence, v u at= + ………... (1) Displacement s = Area under v-t graph ( )1 2s u vt= + ……..(2) Substituting (1) into (2), ( )( )1 2s u u at t= ++ 21 2s ut at= + …...….(3) From (1), vut a −= and substituting into (2), ( ) 221 22 vu v us uv aa −−= +=  22 2v u as= + ………..(4) Before solving any problem involving vector quantities (e.g. s, v, a), there is a need to define the positive direction. Thereafter, all the quantities will take reference to this defined direction for their sign conventions. However, it is possible to define the positive direction opposite to the convention mentioned above whenever it is more convenient to do so. These 4 equations apply only to motion in a straight line with constant acceleration. s = 0 Positive displacement Negative displacement s = 0 Positive displacement Negative displacement

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 7 0 West East displacement 20 km 50 km Example 1 Consider a car moving 50 km from West to East and then 30 km from East to West. Example 2 A car