What’s a Projectile? An object moving in two dimensions under the influence of gravity alone.
You Predict Two identical balls leave the surface of a table at the same time, one essentially dropped the other moving horizontally with a good speed. Which hits the ground first? Courtesy of www.mansfieldct.org/schools/ mms/staff/hand/drop.jpg
Galileo’s Analysis Horizontal and vertical motions can be analyzed separately Ball accelerates downward with uniform acceleration –g Ball moves horizontally with no acceleration So ball with horizontal velocity reaches ground at same time as one merely dropped.
Equations Horizontal v x = v x0 x = x 0 +v x0 t Vertical v y = v y0 –gt y = y 0 + v y0 t – 1/2gt 2 v y 2 = v y0 2 -2gy Assuming y positive up; a x = 0, a y = -g = - 9.80 m/s 2
Problem Solving Read carefully Draw Diagram Choose xy coordinates and origin Analyze horizontal and vertical separately Resolve initial velocity into components List knowns and unknowns Remember v x does not change v y = 0 at top
Examples Horizontal launch Student runs off 10m high cliff at 5 m/s and lands in water. How far from base of cliff? y = -1/2gt 2 (with y up +) time to fall t = (2y/-g) 1/2 = (-20/-9.80) 1/2 = 1.43 s x = v x0 t = 5 x 1.43 = 7.1 m
Extension How fast would the student have to run to clear rocks 10m from the base of the same 10m high cliff? Again t = (2y/-g) 1/2 = 1.43 s x = v x0 t v x0 = x/t = 10m / 1.43s = 7.0 m/s
Upwardly Launched Projectile With Velocity v and Angle v x0 = v 0 cos v y0 = v 0 sin Time in air = 2 v y0 /g Range = R = v x0 t = v 0 cos x 2 v y0 /g = (2v 0 2 /g) sin cos v 0 2 /g )sin2 This is Range Formula Only for y = y 0 R Trigonometric identity: 2 sin cos sin2 Gunner’s version sin2 Rg/v 0 2 v0v0
Questions What is the acceleration vector at maximum height? How would you find the velocity at a given time? How would you find the height at any time? How does the speed at launch compare with that just before impact? 9.80 m/s 2 downward at all times Use kinematics equations for v x and v y, then find v by sqrt sum of squares of them. Use kinematics equation for y Same
Longest Range What angle of launch gives the longest range and why? Assume the projectile returns to the height from which launched. 45 degrees; must maximize sin2 maximum value of sine is one, happens for 90 0 ; then = 45 0 If a launch angle gives a certain range, what other angle will give the same range and why? Hint: R goes as sin cos
Moving Launch Vehicle If ball is launched from moving cart, where will it land?
A Punt Football kicked from 1.00 m above ground at 20.0 m/s at angle above horizontal 0 = 37 0. Find range. Can we use range formula? No. It doesn’t apply since y ≠ y 0 Let x 0 = y 0 = 0 y = -1.00 m Note: Projectile motion is parabolic V y0 = v 0 sin x 12m/s
Punt, continued Find time of flight using y = y 0 + v y0 t – 1/2gt 2 -1.00m = 0 + (12.0m/s)t – (4.90 m/s 2 )t 2 (4.90 m/s 2 )t 2 - (12.0m/s)t – (1.00m) = 0 t = 2.53s or -0.081 s (impossible) x = v x0 t = v 0 cos37 0 t= (16.0 m/s)(2.53s) = 40.5 m
Using Formulas Be sure the formula applies to the situation – that the problem lies within its “range of validity” Make sure you understand what is going on.