A 0.50 kg cue ball makes a glancing blow to a stationary 0.50 kg billiard ball. After the collision the cue ball deflects with a speed of 1.2 m/s at an angle of 30.0° from its original path. Calculate the original speed of the cue ball if the billiard ball ends up travelling at 1.6 m/s.
I think you can safely assume conservation of energy.
1/2 .5 V^2=1/2 .5 1.2^2 + 1/2 .5 1.6^2
solve for V
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Let's assume the initial velocity of the cue ball is V, and the final velocity (after collision) of the cue ball is V'.
2. According to the problem statement, the mass of the cue ball (m1) is equal to the mass of the billiard ball (m2), both equal to 0.50 kg.
3. The initial momentum of the system is given by the equation: momentum_initial = m1 * V + m2 * 0 (since the billiard ball is stationary).
4. The final momentum of the system is given by: momentum_final = m1 * V' * cos(angle) + m2 * velocity_final, where the angle is 30 degrees and the velocity final is given as 1.6 m/s.
Therefore, momentum_final = m1 * V' * cos(30°) + m2 * 1.6.
5. According to the law of conservation of momentum, momentum_initial = momentum_final.
Setting these two equations equal to each other, we have: m1 * V = m1 * V' * cos(30°) + m2 * 1.6.
6. Since m1 = m2 = 0.50 kg, we can substitute these values into the equation: 0.50 * V = 0.50 * V' * cos(30°) + 0.50 * 1.6.
7. Simplifying the equation, we have: 0.50V = 0.50V' * cos(30°) + 0.8.
8. Rearranging the equation: 0.50V - 0.50V' * cos(30°) = 0.8.
9. Plugging in the given value of V' = 1.6 m/s and solving the equation, we get: 0.50V - 0.50 * 1.6 * cos(30°) = 0.8.
10. Using the fact that cos(30°) = √3 / 2, we can simplify the equation further: 0.50V - 0.50 * 1.6 * √3 / 2 = 0.8.
11. Calculating the values in the equation: 0.50V - 0.50 * 1.6 * (√3 / 2) = 0.8.
12. Finally, solving for V: V = (0.8 + 0.50 * 0.50 * 1.6 * (√3 / 2)) / 0.50.
Using a calculator, we can find the value of V to be approximately 2.84 m/s.