Yesterday we looked at what happens to a projectile when it is launched horizontally. However, more generally a projectile could be launched at any angle. Below shows what we need to do to solve problems that involve projectiles launched at an angle to the horizontal. The PowerPoint can also be found here with an example problem to be solved. At the end of the PowerPoint is a task to produce a scale diagram of a projectile's trajectory (i.e. the path that it takes). Any projectile that moves under the influence of gravity will show a parabolic path. This is where the y-displacement is proportional to the square of the x-displacement. You can calculate the values for the x-displacement using the simple equation: dist. = speed x time. The vertical displacement values can be found using the equation s = 1/2 at^2. Where a = -10m/s^2 (for ease of calculation!). Below is a diagram of the graph produced in excel.
Relative motion questions can be found here and are shown below: Projectile Motion
A projectile is an object that once launched continues to move without assistance and only under that influence of gravity. Independence of Horizontal and Vertical Motion. An object launched with some initial horizontal velocity will fall at the same rate as one without any horizontal velocity and the two objects will hit the ground at the same time. This idea is know as the independence of horizontal and vertical motion. The horizontal velocity does not affect the vertical velocity in any way and conversely the vertical acceleration does not affect the horizontal velocity. Ignoring air-resistance we find: 1. The horizontal component of velocity remains constant (because the acceleration in the horizontal direction is zero since there is no component of force acting in the horizontal direction) 2. The vertical component of velocity continues to change with a constant acceleration (due to gravity) NB When solving projectile motion problems we must CONSIDER HORIZONTAL AND VERTICAL MOTION SEPARATELY A projectile with initial horizontal velocity Example: A ball is thrown from the top of a 15 m building with an initial speed of 3 m/s in the horizontal direction. Calculate: a) the time it takes for the ball to hit the ground; b) the distance from the base of the building that the ball lands. a) To work out the time we will consider vertical motion: in the vertical direction we know, u = 0m/s; a = 9.8 m/s^2; s = 15m; and we want to know t. Therefore we select the equation s = ut + 1/2 at^2. Since u = 0: s = 1/2 at^2 Substituting the values into the equation we find that t = sqrt(2s/a) = 1.75 s b) To work out the distance the ball lands from the base of the building we need to consider horizontal motion: We know the time it takes for the ball to hit the ground and we know the horizontal speed (which remains constant throughout) so we can simply do: distance = speed x time = 3 x 1.75 = 5.25 m Yesterday we gained data for dropping balls. Height was changed and time for ball to drop measured. The relevant equation for this situation comes from one of the equations of motion giving:
s = ut + 1/2 at^2 but since u = 0, we can say that s = 1/2 at^2. If we plot a graph of s (y-axis) against t^2 (x-axis) we should get a straight line graph; the gradient of which will be equal to 1/2 a. The main area of uncertainty in this data comes from human reaction time in starting and stopping the stopclock. This is roughly 0.2s. To reduce this uncertainty, a set-up can be used with an electronic timer. When the ball-bearing is released the timer starts. When the ball-bearing hits the pad at the bottom it opens a switch which stops the electronic timer (this set-up is often used with an electromagnet to hold the steel ball in place at the top) Self-marked h/w questions (see previous post) - worked answers at the bottom of this post.
suvat recap questions found here Investigation: Measuring acceleration of free-fall by dropping balls Worksheet can be found here Key things to remember: Select a good range of heights (at least 6 of each); record raw data to the greatest level of precision you can (i.e. if recording height of 1 metre to the nearest cm, record this as 1.00 m); when calculating data always ensure number of significant figures is the same or one more than the data you are calculating from; repeat readings and take an average for each height to improve reliability; use as much graph paper as possible when plotting graph - select suitable scales to do this. Often referred to as the 'suvat' equations, the equations of motion are a group of equations that allow us to analyse the motion of an object with constant acceleration. Equations are shown below: Derivation of the equations of motion can be found here
When solving problems it is good practice to write down the 3 variables that we know and the one variable that we do not. We then select the correct equation that contains those 4 variables. Example Questions: q1-5 page 67 (OCR AS/A Level Physics A - Mike O'Neill) Introduction to Kinematics
Displacement, speed (instantaneous & average), velocity (instantaneous & average), acceleration Vector and scalar quantities Motion graphs (displacement-time, velocity-time, acceleration-time) Notes on the Kinematics section can be found here (although the stuff on motion graphs is not complete yet!) Notes on vectors can be found here |
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