Basketball game problem- projectile motion

In summary, in order for the basketball to hit the basket at the exact time of the siren without reflecting from the board, it must be thrown at an initial velocity of 14.08 m/s and at an angle of 22.56° from the middle of the court. However, to account for the increase in potential energy as the ball travels to a higher point, new calculations must be done to determine the final velocity and impact angle. One method is to use the formula V_f = V_i + a t and compute new velocities based on the new starting point of 306 cm.
  • #1
mmoadi
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0

Homework Statement



Basketball player throws the ball form the middle of the court one second before the game ends. How much must the initial speed of the ball be and at what angle it must be thrown, if it should hit the basket at the exact time of siren without preceding reflection from board? Length of playground is 26 m, the distance between the floor and the ring is 306 cm, and the ball is thrown from height 206 cm. Neglect the rotation of the ball. At what angle and with what velocity (speed) the ball hits the basket?

Homework Equations



Δx= v0xt
Δy= v0yt – ½(gt²)
v0= sqrt(v0x² + v0y²)
tan α= v0y / v0x

The Attempt at a Solution



What I know:
t= 1 s
L= 26 m
Δx= 13 m
h1= 206 cm
h2= 306 cm
Δy= 100 cm = 1 m

What I’m looking for:
v0= ? m/s (initial velocity)
α = ? ° (launch angle)
vf= ? m/s (final velocity)
β= ? ° (impact angle)

① First, I calculated the v0x = vx: v0x= Δx / t = 13 m/s
② Second, I calculated v0y: v0y= (2Δy + gt²) / 2t= 5.4 m/s
③ I know calculated v0 from the Pythagoras formula: v0= sqrt(v0x² + v0y²)= 14.08 m/s
④ And know I calculated the launching angle α: α= (tan)-¹(v0y / v0x)= 22.56°

What is left know are final velocity and impact angle. Can I just state that final velocity = initial velocity (conservation of energy) and that impact angle = launch angle (symmetry of parabola)? Or, do I have to do new calculations because the basketball ring is higher than the launching height of the ball and the parabola is not symmetrical? If second is the case, I really need help, because I don’t know how to approach it now.

Thank you for helping!
 
Last edited:
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  • #2
You can't say that final velocity = initial velocity (conservation of energy), because
you have to account for the increase in potential energy of the ball.

another option is to use V_f = V_i + a t and compute a new v_x and v_y
 
  • #3


I would like to clarify that the equations used in this attempt at a solution are correct for solving a projectile motion problem. However, there are a few additional considerations that need to be taken into account in order to accurately solve this particular basketball game problem.

Firstly, it is important to note that the initial velocity (v0) and launch angle (α) calculated in this attempt at a solution are for a projectile that reaches the height of the basketball ring (h2) and then falls back down. However, in this problem, the ball needs to hit the basket at the exact time of the siren without preceding reflection from the board. This means that the ball needs to travel a distance (h2 - h1) horizontally before reaching the basket. Therefore, the initial velocity and launch angle need to be adjusted to account for this horizontal displacement.

Secondly, the assumption of neglecting the rotation of the ball may not be accurate in this scenario. The rotation of the ball can affect its trajectory and may need to be taken into consideration when calculating the initial velocity and launch angle.

In order to accurately solve this problem, I would suggest using kinematic equations for projectile motion and taking into account the horizontal displacement and potential rotation of the ball. It may also be helpful to break the problem into smaller parts, such as calculating the initial velocity and launch angle for the horizontal displacement first, and then adjusting for the height difference between the launching point and the basketball ring.

I hope this helps in finding a more accurate solution to the basketball game problem. Good luck!
 

1. What is projectile motion in a basketball game?

Projectile motion is the motion of an object (in this case, a basketball) that is thrown or launched into the air and is subject to the forces of gravity and air resistance. In a basketball game, this is seen when the ball is shot or passed, and it follows a curved trajectory before landing in the basket or being caught by another player.

2. How is projectile motion used in a basketball game?

In a basketball game, projectile motion is used to determine the trajectory of the ball when it is shot or passed. This helps players and coaches calculate the distance and angle needed to make a successful shot or pass.

3. What factors influence projectile motion in a basketball game?

The main factors that influence projectile motion in a basketball game are the initial velocity of the ball, the angle at which it is launched, the force of gravity, and air resistance. Other factors like the height and distance of the players also play a role in determining the trajectory of the ball.

4. How does air resistance affect projectile motion in a basketball game?

Air resistance, also known as drag, can affect the trajectory of a basketball by slowing it down and altering its path. This is because the ball experiences a force in the opposite direction of its motion due to the air molecules colliding with it. The effect of air resistance is more significant for slower-moving objects, which is why it is more noticeable in a basketball game than in other sports like baseball.

5. Can projectile motion be used to predict the outcome of a basketball game?

No, projectile motion alone cannot be used to predict the outcome of a basketball game. While it can help players and coaches determine the trajectory of the ball, there are many other factors that influence the outcome of a game, such as player skill, team strategy, and luck. Projectile motion is just one aspect of the complex dynamics of a basketball game.

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