RI Chap 5 Projectile Motion - Lecture Notes

5 PROJECTILE MOTION H2 Physics 9478 Content Page 5.1 Mass vs Weight 2 5.2 Free Fall 2 5.3 One-Dimensional Motion 4 5.4 Two-Dimensional Motion 4 5.5 Work Done & GPE of a Projectile (without air resistance) 10 5.6 Effects of Air Resistance 12 5.7 Appendix 18 Learning Outcomes Candidates should be able to: (a) describe and use the concept of weight as the force experienced by a mass in a gravitational field. (b) describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction. (c) derive, from the definition of work done by a force, the equation PE mg h∆=∆ for gravitational potential energy changes in a uniform gravitational field (e.g. near the Earth’s surface). (d) recall and use the equation PE mg h∆=∆ to solve problems. (e) describe qualitatively, with reference to forces and energy, the motion of bodies falling in a uniform gravitational field with air resistance, including the phenomenon of terminal velocity.

Page | 2 5.1 Mass vs Weight i.e. it is a measure of the inertia of a body. The same force, when applied to a body of larger mass will cause a smaller change in motion. Weight or the gravitational force on a body is an extrinsic property as it depends not just on the mass of the body, but also the gravitational field strength g at the point where the body is. Mathematically W mg= It is important not to confuse weight with mass. 5.2 Free Fall  Galileo’s Free Fall Experiment Galileo's free fall experiment is one of his most famous contributions to physics. It involved the study of bodies falling freely under the influence of gravity. The purpose of the experiment was to challenge the prevailing Aristotelian belief that heavier bodies fall faster than lighter bodies. Galileo hypothesized that in the absence of air resistance, all bodies would fall at the same rate, regardless of their mass. Mass The mass m of a body is the intrinsic property of a body which resists change in motion. Weight The weight W of a body is the force experienced by a mass in a gravitational field.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 3  Acceleration Since the gravitational force is the only force acting on a falling body, netF mg= By Newton’s 2nd Law, mg ma ag = = Thus, the acceleration of the body is dependent on the gravitational field strength, i.e. ag= . The acceleration of a free-falling object near the surface of the Earth has an average value of 29.81 m s− . This is known as the acceleration of free fall. Professor Brian Cox visits NASA’s Space Power Facility in Ohio to see what happens when a bowling ball and a feather are dropped together in the world’s biggest vacuum.  Free Falling Bodies in a Uniform Gravitational Field An object moving freely near the surface of the Earth in the absence of air resistance is said to be undergoing free fall. In such a case, whether the body is moving upwards or downwards, it experiences a constant acceleration directed downwards (towards centre of the Earth) with magnitude of 29.81 m sg −= . For illustration, let us consider the motion of a ball that is projected vertically upwards and falling back to its original position and beyond. Upwards is defined as the positive direction and the reference point where displacement is zero is set at point A. Fig. 5.1 shows the sign of the displacement, velocity and acceleration vectors of the ball at the various points along its motion. displacement velocity acceleration A zero + – B + + – C + zero – D + – – E zero – – F – – – Fig. 5.1 A B C D E F

Page | 4 5.3 One-Dimensional Motion The four kinematic equations that were introduced in Chapter 3 are as follows: • v u at= + • ( )1 2s u vt= + • 21 2s ut at= + • 22 2v u as= + Note that these four equations apply only to motion in a straight line with constant acceleration as shown in Fig. 5.2. Fig 5.2 In the absence of air resistance, bodies moving in air (i.e., free falling) experience a constant vertical acceleration ( 29.81 m syag −= = ) directed downwards, and no horizontal acceleration ( 0xa = ). 5.4 Two-Dimensional Motion  Combining Horizontal & Vertical Motion A body that is projected near the surface of the Earth, such as a kicked football or a batted baseball, describes a curved path in a vertical plane. This kind of motion is called projectile motion. Projectile motion involves motion in the horizontal and vertical directions simultaneously. These perpendicular components of motion are independent yet coordinated, with the vertical motion being similar to an object in free fall.  Important Terms When describing a projectile, the following terms are often used: • Trajectory – the path described by a body, which for this chapter is parabolic. • Range – the horizontal displacement between the point of projection and the point of impact. • Angle of projection – the angle between the direction of projection and the horizontal plane through the point of projection. • Time of flight – time taken from the point of projection to the point of impact

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 5  Key Characteristics of Projectile Motion 1) The trajectory of the projectile is symmetrical about the vertical axis through the highest point. 2) The time the body takes to go from 0ys = to the highest point equals to the time the body takes to go from the highest point to 0ys = because acceleration is constant at g throughout. 3) The path of a projectile is always a parabola.  General Analysis of the Trajectory of a Projectile Projectile motion can be analysed in the following steps: 1) Resolving the displacement , velocity, and acceleration vectors into their horizontal and vertical components (or two perpendicular components). 2) Applying the kinematic equations along the horizontal and vertical directions separately. 3) The resultant velocity at any point can be determined by doing a vector addition of the horizontal and vertical components of the velocity at that point.  Applying Kinematic Equations Consider a cannon ball shot out of a tank at an angle of 30° to the horizontal at a velocity of 1100 m s− , where air resistance is negligible. The path of the cannon ball would be a parabola shown in Fig. 5.5. Horizontal Motion If we ar e looking down from above, we would observe the cannon ball to be moving in a straight line. Neglecting air resistance, the cannon ball moves in a straight line with constant velocity as the acceleration in the horizontal direction is zero. Fi g 5.3 Kinematics Equations 0 x xx xx a uv s ut = = =    tank cannon ball trajectory/path horizontal displacement sx ux vx target 30°

Page | 6 Vertical Motion If we are looking from the front view of the tank, we would observe the cannon ball slowing down while moving upwards until it stops momentarily in the vertical direction, then speeding up while moving downwards. The magnitude of its acceleration is the acceleration of free fall g where 29.81 m sg −= and is directed downwards. Fig 5.4 Kinematics Equations (taking up as positive) ( ) ( ) ( ) 2 2 2 22 2 9.81 m s 9.81 1 2 1 9.812 2 2 9.81 y yyy y yy y y y y yy yy ag v u at ut s ut at ut t v u as us −= −= − = + = +− = + = +− = + =+−     * note that x x st u= Combining Horizontal & Vertical Motions The components of the initial velocity u of the cannon ball are given by 1 1 cos 100cos30 86.6 m s sin 100sin30 50.0 m s x y uu uu θ θ − − = = °= = = °= The velocity v of the cannon ball at any time in the motion can be described as • having a magnitude of 22 xyv vv= + where cosxvv θ= and sinyvv θ= , and • travelling at an angle θ to the horizontal where 1tan y