2: Motion in a Straight Line

1. The DISPLACEMENT, Dx is the change from one position to another, i.e., Dx= x2-x1. Positive values of Dx represent motion in the positive direction (increasing values of x, i.e. left to right looking into the page), while negative values correspond to decreasing x. x = -3 -2 -1 0 1 2 3 2: Motion in a Straight Line Position and Displacement. To locate the position of an object we need to define this RELATIVE to some fixed REFERENCE POINT, which is often called the ORIGIN (x=0). In the one dimensional case (i.e. a straight line), the origin lies in the middle of an AXIS (usually denoted as the ‘x’-axis) which is marked in units of length. Note that we can also define NEGATIVE co-ordinates too. Displacement is a VECTOR quantity. Both its size (or ‘magnitude’) AND direction (i.e. whether positive or negative) are important. REGAN PHY34210

2. Average Speed and Average Velocity We can describe the position of an object as it moves (i.e. as a function of time) by plotting the x-position of the object (Armadillo!) at different time intervals on an (x , t) plot. The average SPEED is simply the total distance travelled (independent of the direction or travel) divided by the time taken. Note speed is a SCALAR quantity, i.e., only its magnitude is important (not its direction). From HRW p15 REGAN PHY34210

3. The average VELOCITY is defined by the displacement (Dx) divided by the time taken for this displacement to occur (Dt). The SLOPE of the (x,t) plot gives average VELOCITY. Like displacement, velocity is a VECTOR with the same sign as the displacement. The INSTANTANEOUS VELOCITY is the velocity at a specific moment in time, calculated by making Dt infinitely small (i.e., calculus!) REGAN PHY34210

4. The instantaneous ACCELERATION is given by a, where, SI unit of acceleration is metres per second squared (m/s2) Acceleration HRW p18 Acceleration is a change in velocity (Dv) in a given time (Dt). The average acceleration, aav, is given by REGAN PHY34210

5. By making the assumption that the acceleration is a constant, we can derive a set of equations in terms of the following quantities Constant Acceleration and the Equations of Motion For some types of motion (e.g., free fall under gravity) the acceleration is approximately constant, i.e., if v0 is the velocity at time t=0, then Usually in a given problem, three of these quantities are given and from these, one can calculate the other two from the following equations of motion. REGAN PHY34210

6. Equations of Motion (for constant a). REGAN PHY34210

7. Alternative Derivations (by Calculus) REGAN PHY34210

8. Free-Fall Acceleration At the surface of the earth, neglecting any effect due to air resistance on the velocity, all objects accelerate towards the centre of earth with the same constant value of acceleration. This is called FREE-FALL ACCELERATION, or ACCELERATION DUE TO GRAVITY, g. At the surface of the earth, the magnitude of g = 9.8 ms-2 Note that for free-fall, the equations of motion are in the y-direction (i.e., up and down), rather than in the x direction (left to right). Note that the acceleration due to gravity is always towards the centre of the earth, i.e. in the negative direction, a= -g = -9.8 ms-2 REGAN PHY34210

9. (a) since a= -g = -9.8ms-2, initial position is y0=0 and at the max. height vm a x=0 Therefore, time to max height from (b) Example A man throws a ball upwards with an initial velocity of 12ms-1. (a) how long does it take the ball to reach its maximum height ? (b) what’s the ball’s maximum height ? REGAN PHY34210

10. ( c) How long does the ball take to reach a point 5m above its initial release point ? Note that there are TWO SOLUTIONS here (two different ‘roots’ to the quadratic equation). This reflects that the ball passes the same point on both the way up and again on the way back down. REGAN PHY34210