Vertical Launch Physics

Let’s use kinematics to analyze an object that has been launched straight up moving 29.4 m/s

Velocity The object will lose 9.8 m/s of velocity every second because of gravity. (We can refer to them as “gravity bites.”)

Position Use kinematics equation 2 to calculate the position of the object at any time. ∆ x = ½ at2 + v0t

9.8 – 9.8 v3 = x3 = 0 m/s 44.1 m ∆ x = ½ at2 + v0t = ½ (-9.8)(3)2 + 29.4(3) v2 = x2 = v1 = x1 = v0 = x0 = 9.8 m/s 39.2 m 19.6 m/s 24.5 m 29.4 m/s 0 m v4 = x4 = v5 = x5 = v6 = x6 = -9.8 m/s 39.2 m -19.6 m/s 24.5 m -29.4 m/s 0 m 0 – 9.8 19.6 – 9.8 ∆ x = ½ at2 + v0t = ½ (-9.8)(2)2 + 29.4(2) ∆ x = ½ at2 + v0t = ½ (-9.8)(4)2 + 29.4(4) -9.8 – 9.8 29.4 – 9.8 ∆ x = ½ at2 + v0t = ½ (-9.8)(1)2 + 29.4(1) ∆ x = ½ at2 + v0t = ½ (-9.8)(5)2 + 29.4(5) -19.6 – 9.8 ∆ x = ½ at2 + v0t = ½ (-9.8)(6)2 + 29.4(6)

THINGS TO NOTICE: v3 = x3 = 0 m/s 44.1 m v2 = x2 = v1 = x1 = v0 = x0 = Path is symmetric. Time Up = Time Down Launch speed = impact speed The velocity at he highest point = 0 BUT the acceleration is still -9.8 m/s2 (turning counts as acceleration). The position is always positive BUT velocity up = + and velocity down = - v2 = x2 = v1 = x1 = v0 = x0 = 9.8 m/s 39.2 m 19.6 m/s 24.5 m 29.4 m/s 0 m v4 = x4 = v5 = x5 = v6 = x6 = -9.8 m/s 39.2 m -19.6 m/s 24.5 m -29.4 m/s 0 m

Symbolic Alphabet Soup Solving for Time END: ∆x = v0 = v = a = t = 0 m launch - launch - 9.8 m/s2 ttotal Solving for Height TOP: ∆x = v0 = v = a = t = h launch 0 m/s - 9.8 m/s2 tup = ½ of ttotal Use to find ttotal at the END Use to find height at the TOP

NUMERIC EXAMPLE A soccer ball is kicked straight up moving 15 m/s. How high will it go? How much air time will it have? END: ∆x = v0 = v = a = t = 0m 15 m/s - 15 m/s - 9.8 m/s2 want TOP: ∆x = v0 = v = a = t = want 15 m/s 0 m/s -9.8 m/s2 ½ (ttotal) = ½ (3.06) = 1.53 s Use 3rd Equation: ∆x = ½ (v0 +v) t Plug in and solve: ∆x = ½ (15 +0) (1.53) = 11.5 m Use 1st Equation: v = at + v0 Solve for t: t = v – v0 a = -15 – 15 -9.8 = 3.06 s

PRACTICE What launch speed is needed to keep a tennis ball in the air for 3.4 seconds if it is hit straight up?

PRACTICE How long will an arrow stay in the air if it is fired straight up going 52 m/s?

PRACTICE How high up will a jumper go if they leave the ground moving 6.4 m/s?

PRACTICE Bob is trying to throw his backpack over a tall fence. If the fence is 7 meters tall, then how much time will the backpack be in the air?

PRACTICE How much launch velocity is needed in order for gymnast to jump 1.2 meters while doing a back flip?

PRACTICE A rocket is launched in the soccer field. After 8 seconds, the rocket crashes back into the grass. How high did the rocket go?