In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in the figure. Assume that the bowling ball, initially traveling at 15.0 m/s, has four times the mass of a pin and that the pin goes off at 75° from the original direction of the ball.
(a) Calculate the speed of the pin.
(b) Calculate the speed of the ball just after collision.
(c) Calculate the angle è through which the ball was deflected. Assume the collision is elastic and ignore any spin of the ball.
To solve this problem, we will use principles of conservation of momentum and conservation of energy.
(a) To calculate the speed of the pin after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
Before the collision, the only object in motion is the ball. Let's define the mass of the ball as m and the mass of the pin as m_pin. Given that the ball has 4 times the mass of the pin, we can say that m = 4m_pin.
The initial velocity of the ball is 15.0 m/s. Since the momentum is the product of mass and velocity, the momentum of the ball before the collision is given by p_initial = m * v_initial. Therefore, p_initial = (4m_pin) * 15.0 m/s.
After the collision, the ball moves off at an angle of 75° from the original direction of the ball. This means that the pin moves off at the same angle, but in the opposite direction.
Let's assume the speed of the pin after the collision is v_pin. We can calculate the momentum of the pin after the collision as p_pin = m_pin * v_pin.
Since momentum is conserved, we can write the conservation equation as:
p_initial = p_pin
(4m_pin) * 15.0 m/s = m_pin * v_pin
Rearranging the equation, we get:
v_pin = (4/15) * 15.0 m/s
Simplifying, we find:
v_pin = 4.0 m/s
Therefore, the speed of the pin after the collision is 4.0 m/s.
(b) To calculate the speed of the ball just after the collision, we need to use the conservation of energy. In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Before the collision, the only object in motion is the ball. The kinetic energy of the ball before the collision is given by KE_initial = (1/2) * m * v_initial^2.
After the collision, the ball moves off at an angle of 75°. Let's assume the speed of the ball after the collision is v_final. The kinetic energy of the ball after the collision is given by KE_final = (1/2) * m * v_final^2.
Using the conservation of energy, we can write the equation:
KE_initial = KE_final
(1/2) * m * v_initial^2 = (1/2) * m * v_final^2
Canceling out the common terms, we get:
v_initial^2 = v_final^2
Taking the square root of both sides, we find:
v_initial = v_final
Therefore, the speed of the ball just after the collision is also 15.0 m/s.
(c) To calculate the angle è through which the ball was deflected, we can use trigonometry.
Since the pin moves off at an angle of 75° in the opposite direction, the angle of deflection for the ball is given by:
è = 180° - 75°
è = 105°
Therefore, the angle through which the ball was deflected is 105°.
In summary:
(a) The speed of the pin after the collision is 4.0 m/s.
(b) The speed of the ball just after the collision is 15.0 m/s.
(c) The angle through which the ball was deflected is 105°.