Question

Try the following:

• get some stuff:
• • a small ball (or some kind of object that will roll - a golf ball or marble or toy car is great, but an empty soup can will do in a pinch)
  • get a tape measure (a yardstick or a ruler will also work. You can also stretch a piece of string and mark off ruler lengths on the string to get the total length.)
  • around ten coins
• Measure the distance from a tabletop or kitchen countertop down to the floor. Record the height in meters. (If you measured the height in inches then convert to meters by dividing the height by 39.36)
• Calculate the time it would take any object to fall from the edge of the tabletop to the floor. Use the y-direction displacement formula: y = v_yot + 1/2 a_y t² where
  • y = the height you measured DOWN to the ground
  • v_yo = the initial vertical velocity - should be zero since an object that rolls off the tabletop will not initially be moving up or down, but only sideways
  • a_y = the acceleration of gravity DOWN = 9.8 m/s²)
  • t = the time

Your Answer:Question 1 options:

Answer

Question 2 (1 point)

Try the following:

• place the small ball on the tabletop, a foot or so from the edge
• get one of the coins
• give the ball a small push so that it rolls off the edge of the table and place the coin about where the ball lands on the floor
• Roll the ball off the tabletop again, this time giving a more forceful push so the ball has more horizontal velocity and again mark its landing spot with a coin
• repeat pushing the ball off the table and marking its landing position several times, each time with a little more force so as to give the ball a higher horizontal velocity when it leaves the tabletop

What is true of each recorded fall? MARK ALL THAT APPLY!

Question 2 options:

	A) No matter how fast the ball leaves the table horizontally, it still takes the same amount of time to fall from the tabletop to the floor
	B) Even when the ball leaves the tabletop with a higher horizontal velocity it always travels the same distance in the x-direction
	C) When the ball leaves the tabletop with a higher horizontal velocity it travels farther in the x-direction

Question 3 (1 point)

Try the following:

• gather all the coins off the floor
• On top of the table, make a small ramp out of a thin board or a magazine. Have the bottom edge of the ramp directly on the table about a foot from the edge. Use a couple books to support the top of the ramp closer to the middle of the table
• Place a ball at the top of the ramp and allow it to roll down the ramp, across the foot of tabletop to the edge and go over the edge. Mark where the ball lands with a coin
• Without changing the angle or position of the ramp, repeatedly release the ball from the top of the ramp and mark each landing spot with a coin

What is true of each recorded fall? MARK ALL THAT APPLY!

Question 3 options:

	A) If all factors could be perfectly controlled the ball would hit in the same spot everytime
	B) The landing spots are pretty uniform and the coins are very closely grouped on the floor.
	C) Everytime a ball rolls off the table, the table itself gets a little bit taller

Question 4 (1 point)

• Go to https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html
• Click the first square that reads "Intro"
• You should see a cannon

Just to make sure you're in the right place, what color is the cannon?

Question 4 options:

	A) blue and yellow
	B) red and blue
	C) shades of gray
	D) red and yellow

Question 5 (1 point)

With all the original default settings (height = 10 m, angle = 0 degrees, vo = 15 m/s), press the red "Fire" button to shoot the cannon. There are three ways to measure the distance the projectile travels in the x-direction. any of the following will work:

• click and drag the red and white target on the ground over to the landing spot so that the center of the target is at the end of the trajectory
• click and drag the tape measure from the toolbox in the top right of the screen. Place the flat leading edge of the tape measure box on the ground directly below the "+" of the cannon and drag the end of the tape to the impact spot
• click and drag the blue time/range/height tool from the toolbox in the top right of the screen. Place the crosshairs over the impact spot and a small yellow dot should appear. This is probably the best tool, since you can also read the time the projectile spent in the air and the range and the height above the ground.

Which of the following is closest to the actual distance the projectile travels in the x-direction?

Question 5 options:

	A) 23.8 m
	B) 21.4 m
	C) 19.2 m
	D) 15.0 m

Question 6 (1 point)

• Find the time that it takes a stone to fall 13 m by using the y-displacement formula.
• Set the cannon at 13 m and fire horizontally and use the blue time/range/height gauge to fine the time

How much time does it take to fall 13 m?

Question 6 options:

	A) 15.96 s
	B) 5.8 s
	C) 1.63 s
	D) 0.90 s

Question 7 (1 point)

If a cannon shoots a ball horizontally at 8 m/s, and the ball starts out 11 m above the floor, how far away from the gun will the ball land?

• Solve the formula using the y-displacement formula to find the time and the x-displacement formula to find the range
• Sent the cannon height at 11m and the initial speed at 8 m/s. Use the blue time/range/height gauge to find the range

What is the horizontal distance the projectile travels?

Question 7 options:

	A) 5.2 m
	B) 12 m
	C) 2.67 m
	D) 15 m

Question 8 (1 point)

Predict the horizontal distance a cannon on a 15 m cliff will shoot if it's horizontal initial velocity is 140 m/s .

Question 8 options:

	A) 207 m
	B) 245 m
	C) 1052 m
	D) 1089 m

Angle Shot

When the cannon is not horizontal, but is aimed either upwards or downwards, several additional considerations have to be made.

• we have to find the x and y components of the inital velocity
  • v_ox = v_o cos θ
  • v_oy = v_o sin θ
• we have to get the signs of the y components of velocity and acceleration and displacement correct. If the gun is angled up and gravity pulls down and the displacement is down. You can pick the y-coordinate sysytem either positive up/negative down OR positive down/negative up. It makes no difference which way you pick, but once you pick everything has to conform to that coordinate system. If we arbitrarily pick positive up
  • the upward initial velocity, v_oy would be up, so it would be a positive number
  • the gravity, a_y pulls down, so it would be a negative number
  • the rock falls down, so the vertical displacement, y points down, so it would be a negative number
• When we use the y-displacement formula to find time, since v_oy is no longer zero, as it was in the horizontal shot, we can't cancel the first right hand term. That means we have to solve the quadratic. Typically using the quadratic formula is the best choice for doing that. The quadratic formula gives two possible solutions - choose the one that makes sense (the positive answer)

For example: A cannonball leaves a cannon at 15 m/s from 10 m above ground, fired at an upward 30 degree angle. Find the time to hit the ground and the horizontal range.

Step 1: Y-direction to find Time	Step 2: X-direction to find Range
voy = vo sin θ = 15 sin 30 = 7.5 (positive since aimed UP) ay = -9.8 m/s^2 (negative since pulls DOWN) y = -15 m (negative since DOWN) t = the time we're looking for y = voy t + 1/2 ayt2 -15 = 7.5 t + 1/2 (-9.8) t2 0 = -4.9 t2 + 7.5 t+15 Get the coefficients for the quadratic formula a = -4.9 , b = 7.5 , c = 15 t = (-b ± √(b2 - 4 a c)) / (2 a ) t = (-7.5 ± √(7.52 - 4(-4.9)(15))) / (2 (-4.9)) t = -1.14s OR 2.67 s	vox = vo cos θ = 15 cos 30 = 13 ax = 0 (gravity doesn't pull sideways, no air resistance) t = whatever we got from Step 1 = 2.67 s x = the range we're looking for x = vox t + 1/2 axt2 x = (13 m/s)(2.67 s) + 1/2 (0) (2.67)2 x = 34.8 m

Question 9

Set up the cannon so that

• the height is 10 m
• initial speed is 15 m/s
• the angle is 0 degrees, horizontal
• fire the cannon!

Tilt the barrel upwards to 30 degrees and FIRE!

Now, tilt the barrel downwards to 30 degrees and - 3, 2, 1 - FIRE!!!

You should see all three paths that the projectile took. You might need the blue measuring device, but MARK ALL THE APPLY THINGS:

Question 9

	A) When you tilt the gun higher, the shot spends less time in the air
	B) More angle = more time
	C) More time = more range
	D) Since the initial velocity and the height is the same for all three shots, the time in the air is the same.

Question 10

A cannonball leaves a cannon at 7 m/s from 4 m above ground, fired at an upward 25 degree angle. Find the time to hit the ground.

Do this by solving the quadratic and by shooting the virtual cannon

Your Answer:

Question 11

A cannonball leaves a cannon at 15 m/s from 5 m above ground, fired at an upward 30 degree angle. Find the horizontal diistance the shot travels.

Do this by solving the quadratic and by shooting the virtual cannon to compare.

Your Answer:

Answer 1