3.4 Projectile Motion

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1 3.4 Projectile Motion

2 Projectile Motion A projectile is anything launched, shot or thrown---i.e. not self-propelled. Examples: a golf ball as it flies through the air, a kicked soccer ball, a thrown football, a speeding bullet. The path that the projectile follows is called a trajectory.

3 2-D Projectile Motion: Basic Equations Assumptions: ignore air resistance g = 9.80 m/s 2, downward ignore Earth s rotation If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is m/s 2

4 2-D Projectile Motion Where g = 9.80 m/s 2. x f = x i + v xi t y f = y i + v yi t 1 2 gt2 v xf = v xi v yf = v yi gt This motion is for a passive projectile. A missile or vehicle (e.g. airplane) may also have self-controlled acceleration (a x, a y, a z ).

5 2-D Projectile Motion x = x i + v x t y = y i + v yi t 1 2 gt2 Motion in the x-y plane will be parabolic, because horizontal position is linear with time, and vertical position is quadratic. t = x x i v x y = y i + v yi x x i v x 1 2 g y = C + Bx + Ax 2 x x i v x 2

6 2-D Projectile Motion A dropped object and an object launched horizontally will fall at the same rate. Motion in the horizontal and vertical directions are independent. The vertical motion dictates the time of flight. Demonstration!

7 A dropped object keeps up with you (neglecting air resistance)

8 A new extreme sport? (a) Write the horizontal and vertical equations of motion for v 0 = 1.30 m/s. (b) What is the x and y position of the basketball at t = s?

9 Two friends jump at the same time. (a) Who hits the water first? (b) Who hits the water with greater speed?

10 Launch Angle Launch angle: direction of initial velocity with respect to horizontal. This affects the location of the vertex.

11 2-D Projectile Motion If an object is launched at an initial angle of θ 0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

12 Launch Angle For angle θ, v 0x = v 0 cos θ and v 0z = v 0 sin θ. This gives the equations of motion: x f = x i + v 0 cos θ t y f = y i + v 0 sin θ t 1 2 gt2 v x = v 0 cos θ = constant v yf = v 0 sin θ gt

13 Range The range of a projectile is the horizontal distance it travels before landing: Range = x f x i = v 0 cos θ t y f = y i + v 0 sin θ t 1 2 gt2 Solve the y- equation for t, defining y = y f y i t = v 0 sin θ ± v 0 sin θ 2 2g y g Range = v 0 cos θ v 0 sin θ± v 0 sin θ 2 2g y g Let s consider special cases θ = 0 and y = 0.

14 Range: Special Case θ = 0 Range = v g y g = v 0 2h g So to jump a chasm you need v 0 w g 2h

15 Range: Special Case y = 0 Range = v 0 cos θ v 0 sin θ+ v 0 sin θ 2 +0 g v cos θ sin θ = v 0 2 sin 2θ g g = Maximum range occurs when sin 2θ = 1 i.e. θ = 45 then R = v 2 0 g

16 Range: Special Case y = 0: The range is a maximum when θ = 45 :

17 Reality Check Air resistance causes horizontal and vertical velocities to decelerate. Wind also affects the range of a projectile. Spin on a projectile also affects its motion ( lift due to relative velocity + Bernoulli s Principle)

18 Projectiles get their initial velocity from Elasticity thrown objects, bow, slingshot, torsion catapult, toy spring dart gun Rotational motion - sling Gravity trebuchet, ramp (e.g. ski jump) Expanding gas blow gun, cannon, rifle Electromagnetic force rail gun, coil gun Self-powered rockets and guided missiles are not projectiles.

19 Hitting a Target If the rifle is fired directly at the target in a horizontal direction, will the bullet hit the center of the target? Does the bullet fall during its flight?

20 Which of the two trajectories shown will result in a longer time for the ball to reach home plate? a) The higher trajectory. b) The lower trajectory. c) They will take the same time. a) The higher trajectory takes longer. The time of flight is determined by the initial vertical velocity component which also determines the maximum height reached.

21 ConcepTest 4.2 You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: Dropping a Package 1) quickly lag behind the plane while falling 2) remain vertically under the plane while falling 3) move ahead of the plane while falling 4) not fall at all

22 ConcepTest 4.2 You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will: Dropping a Package 1) quickly lag behind the plane while falling 2) remain vertically under the plane while falling 3) move ahead of the plane while falling 4) not fall at all Both the plane and the package have the same horizontal velocity at the moment of release. They will maintain this velocity in the x-direction, so they stay aligned. Follow-up: What would happen if air resistance were present?

23 3.4 Clicker Questions

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