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4.1: Two Dimensional and Projectile Motion

Created by: CK-12

In this chapter, we aim to understand and explain the parabolic motion of a thrown object, known as projectile motion. We can break down any initial velocity vector into its vertical and horizontal components. Motion in one direction is unrelated to motion in perpendicular directions, so we can treat the individual components as separate one-dimensional motion problems using all the equations from the last chapter. Once the object has been thrown, the only acceleration is from gravity, in the vertical direction. The horizontal component of velocity remains unchanged (ignoring air resistance) throughout the flight.

Key Equations

Vertical Direction

\begin{cases}+y & = \text{Upward}\\y(t) & = y_0 +v_{0y} t - \frac{1}{2} g t^2\\v_{y}(t) & = v_{0y} -gt \\{v_y}^2 & = {v_{0y}}^2 - 2g (\Delta y)\\a_y = -g &  = -9.8  \text{m/s}^{2}\end{cases}

Horizontal Direction

\begin{cases}x(t) & = x_{0} +v_{0x}t\\v_{x}(t) & = v_{0x} \\a_x & = 0\end{cases}

Note: the initial velocity  v_0 can be separated into  v_{0x} = v_0 \cos{\theta} and  v_{0y} = v_0 \sin{\theta} , where  \theta is the angle between the velocity vector and the horizontal.

Key Concepts

  • In projectile motion, the horizontal displacement of an object from its starting point is called its range.
  • Vertical speed is zero only at the highest point of a thrown object's flight.
  • To work these problems, separate the “Big Three” equations into two sets: one for the vertical direction, and one for the horizontal. Keep them separate.
  • The only variable that can go into both sets of equations is time. You use time to communicate between the two directions.
  • Since in the absence of air resistance there is no acceleration in the horizontal direction, this component of velocity does not change over time. This is a counter-intuitive notion for many. (Air resistance will cause velocity to decrease slightly or significantly depending on the object. But this factor is ignored for the time being.)
  • Motion in the vertical direction must include the acceleration due to gravity, and therefore the velocity in the vertical direction changes over time.
  • The shape of the path of an object undergoing projectile motion in two dimensions is a parabola.
  • We will ignore air resistance in this chapter. Air resistance will tend to shorten the range of the projectile motion by virtue of producing an acceleration opposite to the direction of motion.

Two-Dimensional and Projectile Motion Problem Set

Draw detailed pictures for each problem (putting in all the data, such as initial velocity, time, etc.), and write down your questions when you get stuck.

  1. Determine which of the following is in projectile motion. Remember that “projectile motion” means that gravity is the only means of acceleration for the object.
    1. A jet airplane during takeoff.
    2. A baseball during a Barry Bonds home run.
    3. A spacecraft just after all the rockets turn off in Earth orbit.
    4. A basketball thrown towards a basket.
    5. A bullet shot out of a gun.
    6. An inter-continental ballistic missile.
    7. A package dropped out of an airplane as it ascends upward with constant speed.
  2. Decide if each of the statements below is True or False. Then, explain your reasoning.
    1. At a projectile’s highest point, its velocity is zero.
    2. At a projectile’s highest point, its acceleration is zero.
    3. The rate of change of the x position is changing with time along the projectile path.
    4. The rate of change of the y position is changing with time along the projectile path.
    5. Suppose that after 2 \;\mathrm{s}, an object has traveled 2 \;\mathrm{m} in the horizontal direction. If the object is in projectile motion, it must travel 2 \;\mathrm{m} in the vertical direction as well.
    6. Suppose a hunter fires his gun. Suppose as well that as the bullet flies out horizontally and undergoes projectile motion, the shell for the bullet falls directly downward. Then, the shell hits the ground before the bullet.
  3. Imagine the path of a soccer ball in projectile motion. Which of the following is true at the highest point in its flight?
    1.  v_x = 0, v_y = 0, a_x =0, a_y = 0.
    2.  v_x > 0, v_y = 0, a_x =0, a_y = 0.
    3.  v_x = 0, v_y = 0, a_x =0, a_y = -9.8\;\mathrm{m/s}^2.
    4.  v_x > 0, v_y = 0, a_x =0, a_y = -9.8\;\mathrm{m/s}^2.
    5.  v_x = 0, v_y = 0, a_x =0, a_y = -9.8\;\mathrm{m/s}^2.
  4. A hunter with an air blaster gun is preparing to shoot at a monkey hanging from a tree. He is pointing his gun directly at the monkey. The monkey’s got to think quickly! What is the monkey’s best chance to avoid being smacked by the rubber ball?
    1. The monkey should stay right where he is: the bullet will pass beneath him due to gravity.
    2. The monkey should let go when the hunter fires. Since the gun is pointing right at him, he can avoid getting hit by falling to the ground.
    3. The monkey should stay right where he is: the bullet will sail above him since its vertical velocity increases by 9.8 \;\mathrm{m/s} every second of flight.
    4. The monkey should let go when the hunter fires. He will fall faster than the bullet due to his greater mass, and it will fly over his head.
  5. You are riding your bike in a straight line with a speed of 10 \;\mathrm{m/s}. You accidentally drop your calculator out of your backpack from a height of 2.0 \;\mathrm{m} above the ground. When it hits the ground, where is the calculator in relation to the position of your backpack? (Neglect air resistance.)
    1. You and your backpack are 6.3\;\mathrm{m} ahead of the calculator.
    2. You and your backpack are directly above the calculator.
    3. You and your backpack are 6.3 \;\mathrm{m} behind the calculator.
    4. None of the above.
  6. A ball of mass  m is moving horizontally with speed  v_0 off a cliff of height  h , as shown. How much time does it take the rock to travel from the edge of the cliff to the ground?
    1.  \sqrt{hv_0} .
    2.  \frac{h}{v_0} .
    3.  \frac{hv_0}{g} .
    4.  \frac{2h}{g} .
    5.  \sqrt{\frac{2h}{g}} .
  7. Find the missing legs or angles of the triangles shown.
  8. Draw in the x- and y-velocity components for each dot along the path of the cannonball. The first one is done for you.
  9. A stone is thrown horizontally at a speed of 8.0 \;\mathrm{m/s} from the edge of a cliff 80 \;\mathrm{m} in height. How far from the base of the cliff will the stone strike the ground?
  10. A toy truck moves off the edge of a table that is 1.25 \;\mathrm{m} high and lands 0.40 \;\mathrm{m} from the base of the table.
    1. How much time passed between the moment the car left the table and the moment it hit the floor?
    2. What was the horizontal velocity of the car when it hit the ground?
  11. A hawk in level flight 135\;\mathrm{m} above the ground drops the fish it caught. If the hawk’s horizontal speed is 20.0 \;\mathrm{m/s}, how far ahead of the drop point will the fish land?
  12. A pistol is fired horizontally toward a target 120\;\mathrm{m} away, but at the same height. The bullet’s velocity is 200 \;\mathrm{m/s}. How long does it take the bullet to get to the target? How far below the target does the bullet hit?
  13. A bird, traveling at 20 \;\mathrm{m/s}, wants to hit a waiter 10 \;\mathrm{m} below with his dropping (see image). In order to hit the waiter, the bird must release his dropping some distance before he is directly overhead. What is this distance?
  14. Joe Nedney of the San Francisco 49ers kicked a field goal with an initial velocity of 20 \;\mathrm{m/s} at an angle of 60^\circ.
    1. How long is the ball in the air? Hint: you may assume that the ball lands at same height as it starts at.
    2. What are the range and maximum height of the ball?
  15. A racquetball thrown from the ground at an angle of 45^\circ and with a speed of 22.5 \;\mathrm{m/s} lands exactly 2.5 \;\mathrm{s} later on the top of a nearby building. Calculate the horizontal distance it traveled and the height of the building.
  16. Donovan McNabb throws a football. He throws it with an initial velocity of 30 \;\mathrm{m/s} at an angle of 25^\circ. How much time passes until the ball travels 35\;\mathrm{m} horizontally? What is the height of the ball after 0.5 seconds? (Assume that, when thrown, the ball is 2 \;\mathrm{m} above the ground.)
  17. Pablo Sandoval throws a baseball with a horizontal component of velocity of 25 \;\mathrm{m/s}. After 2 seconds, the ball is 40 \;\mathrm{m} above the release point. Calculate the horizontal distance it has traveled by this time, its initial vertical component of velocity, and its initial angle of projection. Also, is the ball on the way up or the way down at this moment in time?
  18. Barry Bonds hits a 125\;\mathrm{m}(450') home run that lands in the stands at an altitude 30 \;\mathrm{m} above its starting altitude. Assuming that the ball left the bat at an angle of 45^\circ from the horizontal, calculate how long the ball was in the air.
  19. A golfer can drive a ball with an initial speed of 40.0 \;\mathrm{m/s}. If the tee and the green are separated by 100\;\mathrm{m}, but are on the same level, at what angle should the ball be driven? (Hint: you should use  2\cos{(x)}\sin{(x)} = \sin{(2x)} at some point.)
  20. How long will it take a bullet fired from a cliff at an initial velocity of 700 \;\mathrm{m/s}, at an angle 30^\circ below the horizontal, to reach the ground 200 \;\mathrm{m} below?
  21. A diver in Hawaii is jumping off a cliff 45 \;\mathrm{m} high, but she notices that there is an outcropping of rocks 7 \;\mathrm{m} out at the base. So, she must clear a horizontal distance of 7 \;\mathrm{m} during the dive in order to survive. Assuming the diver jumps horizontally, what is his/her minimum push-off speed?
  22. If Monte Ellis can jump 1.0 \;\mathrm{m} high on Earth, how high can he jump on the moon assuming same initial velocity that he had on Earth (where gravity is 1/6 that of Earth’s gravity)?
  23. James Bond is trying to jump from a helicopter into a speeding Corvette to capture the bad guy. The car is going 30.0 \;\mathrm{m/s} and the helicopter is flying completely horizontally at 100 \;\mathrm{m/s}. The helicopter is 120\;\mathrm{m} above the car and 440 \;\mathrm{m} behind the car. How long must James Bond wait to jump in order to safely make it into the car?
  24. A field goal kicker lines up to kick a 44 yard (40 \;\mathrm{m}) field goal. He kicks it with an initial velocity of 22 \;\mathrm{m/s} at an angle of 55^\circ. The field goal posts are 3 meters high.
    1. Does he make the field goal?
    2. What is the ball’s velocity and direction of motion just as it reaches the field goal post (i.e., after it has traveled 40\;\mathrm{m} in the horizontal direction)?
  25. In a football game a punter kicks the ball a horizontal distance of 43 yards (39 \;\mathrm{m}). On TV, they track the hang time, which reads 3.9 seconds. From this information, calculate the angle and speed at which the ball was kicked. (Note for non-football watchers: the projectile starts and lands at the same height. It goes 43 yards horizontally in a time of 3.9 seconds)

Answers to Selected Problems

  1. .
  2. .
  3. .
  4. .
  5. .
  6. .
    1. 13 \;\mathrm{m}
    2. 41 degrees
    3. v_y = 26 \;\mathrm{m/s}; v_x = 45 \;\mathrm{m/s}
    4. 56 degrees, 14 \;\mathrm{m/s}
  7. .
  8. 32 \;\mathrm{m}
    1. 0.5 \;\mathrm{s}
    2. 0.8 \;\mathrm{m/s}
  9. 104 \;\mathrm{m}
  10. t = 0.60 \;\mathrm{s}, 1.8 \;\mathrm{m} below target
  11. 28 \;\mathrm{m}.
    1. 3.5 \;\mathrm{s}.
    2. 35 \;\mathrm{m}; 15 \;\mathrm{m}
  12. 40 \;\mathrm{m}; 8.5 \;\mathrm{m}
  13. 1.3 seconds, 7.1 meters
  14. 50 \;\mathrm{m}; v_{0y} = 30 \;\mathrm{m/s}; 50^0; on the way up
  15. 4.4 \;\mathrm{s}
  16. 19^\circ
  17. 0.5 \;\mathrm{s}
  18. 2.3 \;\mathrm{m/s}
  19. 6 \;\mathrm{m}
  20. 1.4 seconds
    1. yes
    2. 14 \;\mathrm{m/s} @ 23 degrees from horizontal
  21. 22 \;\mathrm{m/s} @ 62 degrees

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