2D Motion: Freefall & Projectiles

1. 2D Motion: Freefall & Projectiles

2. Freefall Define Freefall Any condition in which the only force affecting your motion is gravity. ProjectileAny object in freefall TrajectoryThe path of a projectile.

3. Terminal velocity • As an object falls, it’s velocity increases. As the velocity increases, the air resistance increases. • Air resistance exists because air molecules collide into a falling body creating an upward force that opposes gravity. • When the force of air resistance balances out the force of the falling body, acceleration becomes zero…or Terminal Velocity

4. Terminal velocity cont… • If an object is thrown upwards, the velocity decreases as it rises, due to air resistance. • When it is at the top of its flight, velocity = 0. • When it starts to fall again in a negative direction, velocity increases again due to gravity.

5. Weight • Do not confuse the issue: • MASS AND WEIGHT ARE NOT THE SAME! • Mass is • It does not change unless the object is physically altered. • Weight is • Weight can change any time the gravitational force on an object changes (on the moon)

6. A feather doesn’t fall like a rock! • If acceleration is constant, then as we increase mass, the force of gravity must increase. • This is consistent with what we have previously stated, mass affects gravitational force. • a = F/m • as mass increases so must Force to keep a 9.8 m/s2 • Experience tells us that a feather will fall to the ground much slower that a rock.

7. a is the acceleration of the object due to gravity and is a constant • g = 9.8 m/s2 • This is important because it means that all objects, regardless of their size or mass, should fall at the same rate

8. Acceleration due to gravity = 9.8 m/s2 • As an object falls, the speed increases by the same amount (about 10 m/s) every second! • The distance it falls increases as a square of the time.

9. Example #1 A brick is dropped from the roof of a building under construction. The brick strikes the ground after 4.85 seconds. What’s the brick’s velocity just before it reaches the ground? We can rearrange the acceleration formula to: Vf = vi + a(t) = 0 m/s + -9.8m/s2 (4.85 s) = 47.5 m/s down

10. Example #1 Cont… A brick is dropped from the roof of a building under construction. The brick strikes the ground after 4.85 seconds. How tall is building? We can rearrange the displacement formula to: D = vi(t) + 1/2at2 = 0 m/s(4.85 s) +1/2 (-9.8 m/s2)(4.85)2 = 115.26 m

11. Example #2 A 0.005 kg coffee cup is dropped from rest from a height h above the floor. The cup falls for 1 s before reaching terminal velocity. What is the filter’s acceleration immediately (at t = 0 s),and after it is dropped? The only force acting on the coffee cup is gravity, so acceleration is -9.8 m/s2

12. Example #2 Cont… A 0.005 kg coffee filter is dropped from rest from a height h above the floor. The filter falls for 1 s before reaching terminal velocity. What is the filter’s acceleration when it reaches terminal velocity? Remember the acceleration becomes zero at terminal velocity because the force of gravity has become balanced by the force of air resistance

13. Example 3 • A .0005 kg c coffee cup is dropped from rest from 6 m above the floor. What is the final velocity? • Vf2 = vi2 + 2a(∆y) = 0 + 2(-9.8m/s2) 6m) = √- 117.6 m/s = -10.8 m/s

14. Example 3 Cont… • A .0005kg coffee cup is dropped from rest a height 6m above the floor. How long does it take for the cup to hit the floor? • t= vf – vi a = -10.8 m/s – 0 m/s -9.8 = 1.1 s

15. Other freefall formulas • V = ag(t) vx = cosᴓ(v1) • d = 1/2agt2 vy = sinᴓ(v1) • Vf = v1 + ag(t) • d = v1(t) + ½ agt2

16. Projectile Motion There are 2 components to look at, the horizontal motion and the vertical motion : The horizontal and vertical motion of the projectile are independent of one another! The HORIZONTAL VELOCITY is CONSTANT (ignore air resistance) The VERTICAL ACCELERATION is always 9.8 m/s2 downward due to the force of gravity. If an object is shot straight out it will still fall towards the Earth as if it were dropped, accelerating at 9.8 m/s2

17. Horizontal velocity vector (remains constant) Vertical velocity vector (steadily increasing)

18. Projectile Motion Since Velocity can be SEPARATED into horizontal and vertical components. Solve vertical and horizontal parts of problem separately. ALL the same equations from linear motion work for projectiles. Time is the only quantity which is the same for both vertical and horizontal motion.

19. Projectile Motion The equation that describes horizontal motion in terms of horizontal speed (vx), horizontal distance (dx) and time of travel (t) is dx = vxt

20. Projectile Motion The equation that describes the vertical motion in terms of distance fallen (dy), vertical accel. (g) and time of travel (t) is dy = ½agt2

21. So we can also derive the following useful Equations: • To calculate time: ∆t = √2∆y ÷ -ag • To calculate horizontal velocity: vx = (√-ag ÷ 2∆y) ∆x

22. Example #3 A stone is thrown horizontally at 25 m/s from the top of a cliff 53 meters high. How long does it take the stone to reach the bottom of the cliff? ∆t = √2∆y ÷ -ag = √2(53m) ÷ -9.8 m/s2 = 3.28 s

23. Example # 3 Cont… A stone is thrown horizontally at 25 m/s from the top of a cliff 53 meters high. How far from the base of the cliff does the stone strike the ground? dx = vxt = (25 m/s) 3.28 s = 82 m

24. Example #4 Cliff divers at Hawaii dive from 65 meter high cliffs. The rocks at the base of the cliff protrude 27 meters beyond the edge of the cliff. What is the minimum horizontal velocity needed to safely clear the protruding rocks? vx = (√-ag ÷ 2∆y) ∆x = [√-9.8 ÷ 2(65m)] (27m) = [√ -9.8 ÷ 130 ] (27m) = √.075 (27m) = .27 (27m) = 7.4 m/s