In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. 9-47. The bowling ball, initially traveling at 13.0 m/s, has four times the mass of a pin and the pin flies off at 80° from the original direction of the ball. Assume the collision is elastic and ignore any spin of the ball.
(a) Calculate the speed of the pin just after collision.
(b) Calculate the speed of the ball just after collision.
(c) Calculate the angle through which the ball was deflected.
(a) Well, in order to calculate the speed of the pin just after the collision, we need to apply some physics. But I must warn you, this answer may cause you to split from laughter!
To calculate the speed of the pin, we can use the conservation of momentum. Since the mass of the bowling ball is four times that of the pin, we can write:
(m_ball * v_ball) + (m_pin * v_pin) = (m_ball * v'_ball) + (m_pin * v'_pin),
where v_ball is the initial velocity of the ball, v_pin is the initial velocity of the pin, v'_ball is the final velocity of the ball, and v'_pin is the final velocity of the pin.
By plugging in the values, we can solve for v'_pin:
(4 * m_pin * 13.0 m/s) + (m_pin * v_pin) = (4 * m_pin * v'_ball) + (m_pin * v'_pin).
Simplifying the equation gives:
52.0 m_pin + v_pin = 4 * m_pin * v'_ball + v'_pin.
Unfortunately, without knowing the values of v_pin, m_pin, v'_ball, and v'_pin, we can't solve this equation. So, I'm afraid we'll have to pin this question down for now.
(b) Similarly, we would need more information to determine the speed of the ball just after the collision. So, I'm sorry to say that this answer will have to wait.
(c) Ah, the angle through which the ball was deflected! Now this question gives us some room for fun. The angle is given as 80° from the original direction of the ball. So, the ball went rogue and decided to take a turn, just like a clunky clown trying to avoid slipping on a banana peel! Let's hope it didn't end up in the gutter.
Once again, the lack of additional information hampers our ability to calculate the precise angles and speeds involved. But don't worry, I'll be here juggling jokes while we wait for more data!
To solve this problem, we can use the conservation of momentum and the conservation of kinetic energy.
(a) To calculate the speed of the pin just after the collision, we can use the conservation of momentum. Since the collision is elastic, the total momentum before the collision should be equal to the total momentum after the collision.
Let's denote the initial speed of the ball as v_b, the initial speed of the pin as v_p, the final speed of the ball as v_b', and the final speed of the pin as v_p'.
According to the conservation of momentum:
m_b * v_b + m_p * v_p = m_b * v_b' + m_p * v_p'
Where:
m_b = mass of the ball
m_p = mass of the pin
Given that the ball has four times the mass of the pin, we can write:
4m_p * v_b + m_p * v_p = 4m_p * v_b' + m_p * v_p'
Simplifying the equation:
4v_b + v_p = 4v_b' + v_p'
Since the pin flies off at 80° from the original direction of the ball, we can assume that the magnitude of the pin's velocity stays the same (v_p' = v_p). Therefore, the equation becomes:
4v_b = 4v_b' + v_p
(b) To calculate the speed of the ball just after the collision, we can use the conservation of kinetic energy. Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.
Let's denote the initial kinetic energy of the ball as KE_b, and the final kinetic energy of the ball as KE_b'.
According to the conservation of kinetic energy:
KE_b = KE_b'
The kinetic energy is given by:
KE = (1/2) * mass * (speed)^2
Using this equation, we can deduce that:
(1/2) * 4m_p * v_b^2 = (1/2) * 4m_p * v_b'^2
Simplifying the equation:
v_b^2 = v_b'^2
Taking the square root of both sides:
v_b = v_b'
This means that the speed of the ball just after the collision is the same as its initial speed (v_b' = v_b).
(c) To calculate the angle through which the ball was deflected, we can use trigonometry. The ball initially travels in a straight line, and after the collision, it is deflected at an angle of 80° from its original direction.
Therefore, the angle through which the ball was deflected is 80°.
To summarize:
(a) The speed of the pin just after the collision is the same as its initial speed (v_p' = v_p).
(b) The speed of the ball just after the collision is the same as its initial speed (v_b' = v_b).
(c) The angle through which the ball was deflected is 80°.
To find the solution to this problem, we will use the laws of conservation of momentum and conservation of kinetic energy.
(a) To calculate the speed of the pin just after the collision, we can use the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
The initial momentum of the ball is given by:
p_initial = m_ball * v_ball_initial
where m_ball is the mass of the ball and v_ball_initial is the initial velocity of the ball.
The initial momentum of the pin is given by:
p_initial = m_pin * v_pin_initial
where m_pin is the mass of the pin and v_pin_initial is the initial velocity of the pin.
Since the collision is elastic, the final momentum of the ball and the pin can be expressed as:
p_final_ball = m_ball * v_ball_final
p_final_pin = m_pin * v_pin_final
According to the problem, the mass of the ball is 4 times the mass of the pin:
m_ball = 4 * m_pin
And the pin flies off at an angle of 80° from the original direction of the ball. This implies that the angle between the velocities of the ball and the pin after the collision is also 80°.
Using the conservation of momentum equation, we can write:
m_ball * v_ball_initial + m_pin * v_pin_initial = m_ball * v_ball_final + m_pin * v_pin_final
Substituting the values we have:
4 * m_pin * 13.0 m/s + m_pin * 0.0 m/s = 4 * m_pin * v_ball_final + m_pin * v_pin_final
Now, let's move on to part (b) to calculate the speed of the ball just after the collision.
(b) Since the collision is elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.
The initial kinetic energy of the ball is given by:
KE_initial_ball = (1/2) * m_ball * (v_ball_initial)^2
The initial kinetic energy of the pin is given by:
KE_initial_pin = (1/2) * m_pin * (v_pin_initial)^2
The final kinetic energy of the ball is given by:
KE_final_ball = (1/2) * m_ball * (v_ball_final)^2
The final kinetic energy of the pin is given by:
KE_final_pin = (1/2) * m_pin * (v_pin_final)^2
Using the conservation of kinetic energy equation, we can write:
KE_initial_ball + KE_initial_pin = KE_final_ball + KE_final_pin
Substituting the values we have:
(1/2) * 4 * m_pin * (13.0 m/s)^2 + (1/2) * m_pin * (0.0 m/s)^2 = (1/2) * 4 * m_pin * (v_ball_final)^2 + (1/2) * m_pin * (v_pin_final)^2
Now, let's move on to part (c) to calculate the angle through which the ball was deflected.
(c) The angle through which the ball was deflected can be found using trigonometry. Since the velocities of the ball and the pin make an angle of 80° after the collision, we can use the sine function to calculate the angle of deflection.
Using the equation:
sin(angle) = (side opposite to the angle) / (hypotenuse)
In this case, the side opposite to the angle is the speed of the pin just after the collision and the hypotenuse is the speed of the ball just after the collision.
Therefore, the angle through which the ball was deflected can be calculated using:
angle = sin^(-1)((speed of the pin just after the collision) / (speed of the ball just after the collision))
By solving these equations simultaneously, we can determine the answers to all three parts of the problem.