VJC 2023 Kinematics LN

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 1 TOPIC 2 KINEMATICS CONTENT 1. Distance and Displacement 2. Speed and Velocity 3. Displacement-Time graphs 4. Acceleration 5. Velocity-Time graphs 6. Free fall in a Gravitational field w/o air resistance 7. Deriving the equations of Kinematics 8. Free fall in a Gravitational field w air resistance 9. Projectile Motion LEARNING OUTCOMES (a) define and use displacement, speed, velocity and acceleration (b) use graphical methods to represent distance, displacement, speed, velocity and acceleration use SI base units to check the homogeneity of physics equations (c) identify and use the physical quantities from the gradients of 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, including the motion of bodies falling in a uniform gravitational field without air resistance (f) describe qualitatively the motion of bodies falling in a uniform gravitational field with air resistance (g) describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction. 0. OVERVIEW OF KINEMATICS To begin learning about classical mechanics, our initial concentration will be solely on an object's movement, disregarding any forces that may impact or alter it. This particular branch of classical mechanics is called kinematics which is the study of motion, a continuos change of position with time. In this chapter we’ll discuss how to quantify that using three key quantities: displacement, velocity and acceleration for objects moving with respect to a coordinate axis in one and two dimensions. 1. DISTANCE AND DISPLACEMENT We start with a few definitions: • Distance is the total length of the path travelled by a body regardless of direction travelled. It is a scalar quantity meaning that it only has magnitude but no direction. • Displacement: the distance travelled in a straight line in a specified direction, from some reference point. It is a vector quantity, meaning that it has both magnitude and direction.

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 2 Example 1: From point A, a boy travels east for 100 m to point B, then travels north for 300 m to point C. He then travels a further 200 m towards the east, reaching point D. What is the total distance he travelled, and what is his final displacement from his starting position? 100 m 300 m 200 m θ A B C D displacement 1.1 Displacement along a straight-line path To define displacement, we fix need a reference starting position which can be chosen arbitrarily. Following which, we decide on a sign convention, e.g. take “to the right” to be positive. Change in displacement of object A = +3 m. Change in displacement of object B = - 4 m. Change in displacement of object C = +6 m. (Final displacement of C is +4 m.)

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 3 2. SPEED AND VELOCITY • Speed is defined as the rate of change of distance travelled or as the distance travelled per unit time. • The use of “per” in the definition is important because it signifies that speed is a ratio: takentime travelledDistanceSpeed= • Speed is a scalar quantity. • Velocity is defined as the rate of change of displacement or as the change of displacement per unit time. Change of displacementVelocity time taken= • Velocity is a vector quantity. • The S.I. units for both speed and velocity are metres per second (m s−1). Example 2: In the previous example, the boy took 100 s to complete the whole journey ABCD. What is his average velocity over the whole journey? Answer: Average velocity = change of displacement time taken 100 m 300 m 200 m θ A B C D 424 m

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 4 2.1 Average velocity vs Instantaneous velocity • Average speed is defined as the total distance travelled over the total time taken. • Average speed = distance travelled time taken = D t ∆ ∆ • Average velocity is defined as the total displacement over the total time taken. • Average velocity = change of displacement time taken = s t ∆ ∆ • Instantaneous speed is defined as the rate of change of distance (with time). • Instantaneous speed= instantaneous rate of change of distance travelled = dD dt • Instantaneous velocity is defined as the rate of change of displacement (with time). • Instantaneous velocity = instantaneous rate of change of displacement = ds dt Example 3: In the previous example, the boy took 100 s to complete the whole journey ABCD. What is his average velocity over the whole journey? A car starts moving from rest along a straight path without reversing, and its distance from its starting point at various times are shown in the table below. time / s 0 1.0 2.0 3.0 4.0 5.0 distance travelled / m 0 5.0 25 68 147 210 Estimate its instantaneous speed at 3.5 s from the start. Answer: Instantaneous speed (estimated) at 3.5 s = Change of displacement between 3 s and 4 s takentime = =

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 5 3. DISPLACEMENT -TIME GRAPH • Graphical depictions provide a comprehensive account of how objects move. • The two primary aspects that receive attention in graphical interpretation are the area and the gradient. • When graphing a dependent variable y over time (with time as the horizontal axis), the gradient signifies the rate at which y changes. The following table summarises the graphs you will learn: Type of graph Gradient represents Area represents Displacement – time Velocity Not Applicable Velocity – time Acceleration Change in Displacement Acceleration-time Not Applicable Change in Velocity Example 4: A body moves at constant speed along a straight-line path, and travels 10 m in a time of 5 s. It then stays at rest for 3 s. After that, it reverses its direction of travel and travels a distance of 8 m in 3 s. Draw a distance-time graph, as well as a displacement-time graph, to illustrate the motion of the body. Answer: 0 5 8 11 t/s D/m 10 18 0 5 8 11 t/s s/m 10 2 distance-time graph displacement-time graph 3.1 Gradient of a displacement-time graph (and distance-time graph) Gradient of a distance-time graph = dD dt = speed Gradient of a displacement-time graph = dt ds = velocity Example 5: In the previous example, calculate the speeds for the three stages of the motion of the body, and draw a speed-time graph for the motion. Similarly, calculate the velocities for the three stages, and draw a velocity-time graph for the motion.

9749 H2 Physics; 8867 H1 Physics Victoria Junior College Lecture Notes Topic 2:Kinematics 6 Answer: From distance-time graph, During the first 5 s, speed = D t ∆ ∆ = From t = 5 s to t = 8 s, speed = From t = 8 s to t = 11 s, speed = From displacement-time graph, For the first 8 s, velocity = speed, since there is no change in direction of travel. From t = 8 s to t = 11 s, velocity = 8 11 10 2 − − = −2.667 ≈ −2.7 m s-1 0 5 8 11 t/s v/m s-1 2 2.7 speed-time graph 0 5 8 11 t/s v/m s-1 2 −2.7 velocity-time graph Example 6: Sketch the velocity-time graph and deduce the corresponding displacement-time graph for each of the following situations. Assume that the body moves along a straight-l