ENGINEERING SCIENCE (CHAPTER 4)

CHAPTER 4 LINEAR MOTION

Introduction • This chapter is a review of the basic terms used in describing linear motion with a derivation of the equations used in tackling problems involving such motion. • Also considered are the graphs of distance time and velocity-time and the data that can be extracted from such graphs. • The vector nature of velocity is considered and its resolution into components, this enabling problems to be tackled which involve projectiles

1.1 Basic terms Following are basic terms used in the description of linear motion: 1. Distance is the distance along the path of an object, whatever the form of the path. Thus, if we say the distance covered in the motion of a car was, say, 3 km then the 3 km could have been covered along a straight road and the car be 3 km away from its start point. Another possibility is, however, that the 3 km was round a circular track and the car at the end of its 3 km might have been back where it started

2. Displacement is the distance in a straight line between the start and end points of some motion. Thus a displacement of 3 km would mean that at the end of the motion that an object was 3 km away from the start point. 3. Speed is the rate at which distance is covered. Thus a car might be stated as having a speed of 50 km/h.

4. Average speed is the distance covered in a time interval divided by the time taken: A car which covers 80 km in 1 hour will have an average speed of 80 km/h over that time. During the hour it may, however, have gone faster than 80 km/h for part of the time and slower than that for some other part.

5. A constant or uniform speed occurs when equal distances are covered in equal intervals of time, however small we consider the time interval. Thus a car with an average speed of 60 km/h for 1 hour will be covering distance at the rate of 60 km/h in the first minute, the second minute, over the first quarter of an hour, over the second half hour, indeed over any time interval in that hour.

6. Velocity is the rate at which displacement along a straight line changes with time. Thus an object having a velocity of 5 m/s means that the object moves along a straight line path at the rate of 5 m/s. 7. Average velocity is the displacement along a straight line occurring in a time interval divided by that time:

Thus an object having a displacement of 3 m along a straight line in a time of 2 s will have an average velocity in the direction of the straight line of 1.5 m/s over that time. During the 2 s there may be times when the object is moving faster or slower than 1.5 m/s. 8. A constant or uniform velocity occurs when equal displacements occur in the same straight line direction in equal intervals of time, however small the time interval. Thus an object with a constant velocity of 5 m/s in a particular direction for a time of 30 s will cover 5 m in the specified direction in each second of its motion.

9. Acceleration is the rate of change of velocity with time. The term retardation is often used to describe a negative acceleration, i.e. when the object is slowing down rather than increasing in velocity. 10. Average acceleration is the change of velocity occurring over a time interval divided by the time:

• Thus if the velocity changes from 2 m/s to 5 m/s in 10 s then the average acceleration over that time is (5 - 2)/10 = 0.3 m/s². If the velocity changes from 5 m/s to 2 m/s in 10 s then the average acceleration over that time is (2 - 5)/10 = -0.3 m/s², i.e. it is a retardation. 11. A constant or uniform acceleration occurs when the velocity changes by equal amounts in equal intervals of time, however small the time interval. Thus an object with a constant acceleration of 5 m/s² in a particular direction for a time of 30 s will change its velocity by 5 m/s in the specified direction in each second of its motion.

1.2 Straight line motion • The equations that are derived in the following discussion all relate to uniformly accelerated motion in a straight line. If u is the initial velocity, i.e. at time t = 0, and v the velocity after some time t, then the change in velocity in the time interval t is (v - U). Hence the acceleration a is (v - u)/ t. Rearranging this gives: V = u + at







1.3 Vectors • Velocity and acceleration are vector quantities (see Chapter 3). A vector quantity is one for which both its magnitude and direction have to be stated for its effects to be determined; they have to be added by methods which take account of their directions, e.g. the parallelogram method.



1.3.1 Resolution into components

1.4 Motion under gravity • All freely falling objects in a vacuum fall with the same uniform acceleration directed towards the surface of the earth as a result of a gravitational force acting between the object and the earth. • This acceleration is termed the acceleration due to gravity g. • For most practical purposes, the acceleration due to gravity at the surface of the earth is taken as being 9.81 m/s². • The equations for motion of a falling object are those for motion in a straight line with the acceleration as g.

• When an object is thrown vertically upwards it suffers an acceleration directed towards the surface of the earth. • An acceleration directed in the opposite direction to which an object is moving is a retardation, i.e. a negative acceleration since it results in a final velocity less than the initial velocity. • The result is that the object slows down. The object slows down until its velocity upwards eventually becomes zero, it then having attained its greatest height above the ground. Then the object reverses the direction of its motion and falls back towards the earth, accelerating as it does; we have then the acceleration of +g

Vertical motion under gravity



1.4.1 Motion down an inclined plane • For free fall the acceleration is g downwards. However, for objects moving down a smooth inclined plane, as in Figure 4.4, vertical motion is not possible. The result is an acceleration down the plane which is due to the resolved component of g in that direction: • acceleration down plane due to gravity = g sin Ѳ



1.5 Graphs of motion This section is a discussion of how graphs can be used to describe the motion of an object.





1.5.2 Velocity-time graphs • If the velocity of an object is measured at different times then a velocity time graph can be drawn. Acceleration is the rate at which the velocity changes. Thus, for the graph shown in Figure 4.9, the velocity changes from v₁ to v₂ when the time changes from t₁ to t₂. Thus the acceleration over that time interval is (v₂ - v₁)/(t₂- t₁). But this is the gradient of the graph. Thus: • acceleration = gradient of the velocity- time graph







Activities



2. Analyse the motion of a car as it accelerates from rest and is changed through the gears from measurements made of the speedometer reading at different times during the motion. From your results obtain velocity-time and speed-time graphs.