See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/342992092 Projectile Motion: A Short Teaching Presentation Presentation · July 2020 DOI: 10.13140/RG.2.2.27881.93288 CITATIONS READS 0 210 1 author: Kamil Walczak City University of New York - Queens College 113 PUBLICATIONS 312 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: "Nonlinear Transport and Noise Properties of Acoustic Phonons" View project "Transport Phenomena in Carbon Nanotube Fibers" View project All content following this page was uploaded by Kamil Walczak on 16 July 2020. The user has requested enhancement of the downloaded file.
Motion in 2-D Space “Projectile Motion” OBJECTIVES ➢ To discuss projectile motion as a combination of a uniform horizontal motion and a uniformly decelerated/accelerated vertical motion. ➢ To justify a few formulas and draw some general conclusions on the basis of kinematic equations; and to verify them via PhET interactive simulations. Dr. Kamil Walczak (Department of Science, Maritime College State University of New York)
Projectile Motion y = 0 y 0 x y 0 x 0 x y m 0 g = 9 . 8 0 y 2 s 0 x Earth’s Surface x 0 x Projectile Motion is a special motion of an object thrown y upward at a certain angle to the horizontal line which is moving along a curved (parabolic) path under the action of the force of gravity only (air resistance is neglected). Kamil Walczak 07 / 16 / 2020
Projectile Motion: Decomposition Projectile Motion may be decomposed into two independent types of motion: (1) the uniform (constant-velocity) straight-line motion along the x-axis and (2) uniformly decelerated/accelerated motion along the y-axis (object is affected only by the downward force of gravity, air resistance is ignored). Initial velocity in the projectile motion: ˆ ˆ = + i j 0 0 x 0 y 0 = 0 y cos( ) 0 x 0 = sin( ) 0 y 0 0 x Kamil Walczak 07 / 16 / 2020
Projectile Motion: Basic Equations Kinematic equations for horizontal motion: = = a x = cos( ) 0 x 0 x 0 = = x 0 = (Eq.1) x t t cos( ) 0 0 x 0 Kinematic equations for vertical motion: = − = − (Eq.2) gt sin( ) gt = − a y g y 0 y 0 1 1 y 0 = = − = − (Eq.3) 2 2 0 y t gt t sin( ) gt 0 y 0 2 2 Kamil Walczak 07 / 16 / 2020
Parabolic Trajectory Parabolic Trajectory of Projectile Motion From Eq.1 – kinematic equation for horizontal position: / cos( ) x 0 = = t x t cos( ) 0 cos( ) 0 From Eq.3 – kinematic equation for vertical position: 1 = − 2 y t sin( ) gt 0 2 g = − − = 2 2 y tan( ) x x Bx Cx y 2 2 2 cos ( ) 0 (Twisted Parabola) Kamil Walczak 07 / 16 / 2020
Maximum Height The Highest Point in Projectile Motion From Eq.2 – kinematic equation for vertical velocity: + gt sin( ) = − = h = 0 sin( ) gt 0 t y 0 h h g / g = = for t t y h h From Eq.3 – kinematic equation for vertical position: 1 1 = − = − 2 2 y t sin( ) gt h t sin( ) gt 0 0 h h 2 2 2 2 2 2 2 2 sin ( ) sin ( ) sin ( ) = − = = 0 0 0 h h g 2 g 2 g Kamil Walczak 07 / 16 / 2020
Maximum Height: Typical Problems An object is thrown vertically upward with an initial velocity 70 m/s. What would be the height (or vertical distance) which this object can reach before start falling down? A stone is projected at the cliff with an initial speed 42 m/s directed at an angle 60 degrees above the horizontal. How long will it take the stone to reach the maximum height above the ground? Kamil Walczak 07 / 16 / 2020
Horizontal Range The Horizontal Range for Projectile Motion The time travel for horizontal range is twice the time travel for maximum height: 2 sin( ) = = 0 t 2 t d h g = = for t t x R d From Eq.1 – kinematic equation for horizontal position: = = R t cos( ) x t cos( ) 0 d 0 2 2 2 sin( ) cos( ) sin( 2 ) = = = = = O O 0 0 for 2 90 45 R R g g (Optimal Angle) Kamil Walczak 07 / 16 / 2020
Horizontal Range: Typical Problems A stone is projected at the cliff with an initial speed 42 m/s directed at an angle 60 degrees above the horizontal. Find the distance at which the stone will hit the ground. An object is thrown upward with an initial speed 35 m/s directed at an angle 50 degrees above the horizontal. How much time will pass until the object hits the ground? Kamil Walczak 07 / 16 / 2020
Height/Range Ratio The Height/Range Ratio in Projectile Motion Horizontal Range: Maximum Height: 2 2 2 2 sin( ) cos( ) sin ( ) = = 0 0 R h g 2 g The ratio of maximum height to horizontal range: 2 2 h sin ( ) g sin( ) = = 0 2 R 2 g 2 sin( ) cos( ) 4 cos( ) 0 h 1 h tan( ) = = = O = for 45 tan( ) 1 R 4 R 4 (Optimal Angle) Kamil Walczak 07 / 16 / 2020
Concluding Remarks Neglecting air resistance, the parabolic trajectory of the object and the time of its travel in the projectile motion are independent of its mass and size! Hypothesis: the trajectory of the object will become asymmetric and the time travel will increase in the presence of air resistance. Neglecting air resistance, the maximum height (the highest point) in the projectile motion is independent of object’s mass and size! Hypothesis: the maximum height will decrease when the analyzed object is affected by air resistance. Neglecting air resistance, the horizontal range is the longest for the optimal angle at which the object is thrown equal to forty-five degrees! Hypothesis: the optimal angle will decrease when air resistance is included. Neglecting air resistance, the horizontal range at optimal angle is four times larger than the maximum height (the highest point) in the projectile motion! Hypothesis: this ratio will change when air resistance is taken into account. Kamil Walczak 07 / 16 / 2020
Drag Force (Air Resistance, Fluid Friction) Drag force is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid (air). The ultimate cause of a drag is viscous friction. Drag force depends on the properties of the fluid, and on the size, shape and speed of the object immersed within the fluid (air). - speed of the object - density of the fluid - drag coefficient - cross-sectional area In the physics of sports, drag force is necessary to explain the performance of basketball players and runners. PhET Simulator: https://phet.colorado.edu/en/simulation/projectile-motion Kamil Walczak 07 / 16 / 2020 View publication stats View publication stats
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