Once a favorite playground sport, dodgeball is becoming increasingly popular among adults of all ages who want to stay in shape and who even form organized leagues. Gutball is a less popular variant of dodgeball in which players are allowed to bring their own (typically nonregulation) equipment and in which cheap shots to the face are permitted. In a gutball tournament against people half his age, a physics professor threw his 0.4-kg soccer ball at a kid throwing a 0.6-kg basketball. The balls collided in midair (see the figure), and the basketball flew off with an energy of 93.1 J at an angle θ = 32.4° relative to its initial path. Before the collision, the energy of the soccer ball was 100 J and the energy of the basketball was 112 J. At what angle and speed did the soccer ball move away from the collision?
angle 1
speed
To find the angle and speed at which the soccer ball moves away from the collision, we will use the principle of conservation of linear momentum and conservation of kinetic energy.
1. Conservation of Linear Momentum:
Before the collision:
Momentum of the soccer ball = mass of soccer ball * velocity of soccer ball
Momentum of the basketball = mass of basketball * velocity of basketball
After the collision:
Momentum of the soccer ball = mass of soccer ball * velocity of soccer ball (after collision)
Momentum of the basketball = mass of basketball * velocity of basketball (after collision)
Since there is no external force acting on the system, the total linear momentum before and after the collision should be conserved:
Momentum of the soccer ball + Momentum of the basketball = Momentum of the soccer ball (after collision) + Momentum of the basketball (after collision)
2. Conservation of Kinetic Energy:
Before the collision:
Kinetic energy of the soccer ball = 0.5 * mass of soccer ball * (velocity of soccer ball)^2
Kinetic energy of the basketball = 0.5 * mass of basketball * (velocity of basketball)^2
After the collision:
Kinetic energy of the soccer ball = 0.5 * mass of soccer ball * (velocity of soccer ball (after collision))^2
Kinetic energy of the basketball = 0.5 * mass of basketball * (velocity of basketball (after collision))^2
Similarly, since there is no external force doing work on the system, the total kinetic energy before and after the collision should be conserved:
Kinetic energy of the soccer ball + Kinetic energy of the basketball = Kinetic energy of the soccer ball (after collision) + Kinetic energy of the basketball (after collision)
Now, let's calculate the velocities of the soccer ball and the basketball after the collision using the conservation principles:
1. Conservation of Linear Momentum:
(mass of soccer ball * velocity of soccer ball) + (mass of basketball * velocity of basketball) = (mass of soccer ball * velocity of soccer ball (after collision)) + (mass of basketball * velocity of basketball (after collision))
Since the mass of the soccer ball and basketball are given as 0.4 kg and 0.6 kg respectively, the equation becomes:
(0.4 kg * velocity of soccer ball) + (0.6 kg * velocity of basketball) = (0.4 kg * velocity of soccer ball (after collision)) + (0.6 kg * velocity of basketball (after collision)) ----(Equation 1)
2. Conservation of Kinetic Energy:
(0.5 * mass of soccer ball * (velocity of soccer ball)^2) + (0.5 * mass of basketball * (velocity of basketball)^2) = (0.5 * mass of soccer ball * (velocity of soccer ball (after collision))^2) + (0.5 * mass of basketball * (velocity of basketball (after collision))^2)
Simplifying this equation using the given information:
(0.5 * 0.4 kg * (velocity of soccer ball)^2) + (0.5 * 0.6 kg * (velocity of basketball)^2) = (0.5 * 0.4 kg * (velocity of soccer ball (after collision))^2) + (0.5 * 0.6 kg * (velocity of basketball (after collision))^2) ----(Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with two unknowns (velocity of soccer ball (after collision) and velocity of basketball (after collision)). We can solve these equations to find the velocities.
Using the given energy values before the collision (100 J for soccer ball and 112 J for basketball), we can substitute the kinetic energy values in Equation 2.
After solving the equations, we can find the velocities of the soccer ball and basketball after the collision. From the velocity of the soccer ball, we can calculate its speed using the equation: speed = magnitude of velocity.
Finally, to find the angle at which the soccer ball moves away from the collision, we need to use the given information of the angle (θ = 32.4°) at which the basketball flew off. The angle of the soccer ball can be calculated using the equation: angle = 180° - angle of basketball.
Please note that this calculation involves solving the equations and using the given values. For an accurate answer, the numerical values need to be substituted into the equations.
To find the angle and speed at which the soccer ball moves away from the collision, we can use the law of conservation of momentum and the law of conservation of kinetic energy.
1. Conservation of momentum:
Before the collision, the total momentum of the system (soccer ball + basketball) is zero, as the balls are not moving in the horizontal direction. Therefore, the momentum after the collision must also be zero.
Let the initial velocity of the soccer ball be v1 and the final velocity after the collision be v2.
The momentum equation can be written as:
(mass of soccer ball)(initial velocity of soccer ball) + (mass of basketball)(initial velocity of basketball) = (mass of soccer ball)(final velocity of soccer ball) + (mass of basketball)(final velocity of basketball)
(0.4 kg)(v1) + (0.6 kg)(0) = (0.4 kg)(v2) + (0.6 kg)(93.1 J)
Simplifying the equation:
0.4v1 = 0.4v2 + 55.86
2. Conservation of kinetic energy:
Before the collision, the total kinetic energy of the system is the sum of the kinetic energies of the soccer ball and the basketball. After the collision, the total kinetic energy is the sum of the kinetic energies of the soccer ball and the basketball.
The kinetic energy equation can be written as:
(1/2)(mass of soccer ball)(initial velocity of soccer ball)^2 + (1/2)(mass of basketball)(initial velocity of basketball)^2 = (1/2)(mass of soccer ball)(final velocity of soccer ball)^2 + (1/2)(mass of basketball)(final velocity of basketball)^2
(1/2)(0.4 kg)(v1)^2 + (1/2)(0.6 kg)(0)^2 = (1/2)(0.4 kg)(v2)^2 + (1/2)(0.6 kg)(93.1)^2
Simplifying the equation:
0.2v1^2 = 0.2v2^2 + 2714.049
Now we have two equations with two unknowns (v1 and v2). We can solve these equations simultaneously to find the values.
Substitute (0.4v2 + 55.86) from the momentum equation into the kinetic energy equation:
0.2v1^2 = 0.2(0.4v2 + 55.86)^2 + 2714.049
Simplify the equation by expanding and collecting terms:
0.2v1^2 = 0.08v2^2 + 4.4872v2 + 3129.4016
Now substitute v2 = (93.1)(cosθ), where θ is the angle of the basketball's final velocity relative to its initial path:
0.2v1^2 = 0.08(93.1cosθ)^2 + 4.4872(93.1cosθ) + 3129.4016
Simplify the equation further:
0.2v1^2 = 6.819667cos^2θ + 418.87812cosθ + 3129.4016
From here, it depends on the numerical values of cosθ to solve the equation for v1. Substituting the cosine values will give you a quadratic equation from which you can solve for v1.
Once you have found v1, you can find the angle at which the soccer ball moves away using the equation v2 = v1*cosθ.
Note: Without the specific numerical values of cosθ given in the problem, it's not possible to calculate the exact angle and speed of the soccer ball after the collision.