A 50 gram ball enters a a pendulum with mass 200 g. The pair then swings up to a height of 10 cm.
Find the velocity at which the pair move immediately after the collision. Your answer is incorrect.
1.4 m/s
Find the initial velocity of the ball before the collision.
To find the initial velocity of the ball before the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant before and after a collision.
Let's denote the initial velocity of the ball as Vb and the initial velocity of the pendulum as Vp. Since the mass of the pendulum is 200 grams and the mass of the ball is 50 grams, we can write the equation for the conservation of momentum as:
(mass of the ball * initial velocity of the ball) + (mass of the pendulum * initial velocity of the pendulum) = (total mass * final velocity)
Using the given values, we have:
(50 g * Vb) + (200 g * Vp) = ((50 g + 200 g) * final velocity)
Now, we need to find the final velocity of the system. We know that the pair swings up to a height of 10 cm, which means that the potential energy of the system is converted into kinetic energy when it reaches its maximum height. At maximum height, the entire energy is kinetic, so we can write the energy equation as:
(mass of the pair * final velocity^2) / 2 = (mass of the pair * g * height)
Plugging in the values for mass of the pair (250 g) and height (10 cm = 0.1 m), we get:
(250 g * final velocity^2) / 2 = (250 g * 9.8 m/s^2 * 0.1 m)
Simplifying the equation:
final velocity^2 = 9.8 m/s^2 * 0.1 m * 2
final velocity^2 = 1.96 m^2/s^2
Taking the square root of both sides:
final velocity = √(1.96 m^2/s^2)
final velocity ≈ 1.4 m/s
Thus, the final velocity of the system immediately after the collision is approximately 1.4 m/s.
Now we can substitute this value back into the conservation of momentum equation to find the initial velocity of the ball:
(50 g * Vb) + (200 g * Vp) = ((50 g + 200 g) * 1.4 m/s)
Now we can solve the equation for Vb:
50 g * Vb + 200 g * Vp = 250 g * 1.4 m/s
50 g * Vb = 250 g * 1.4 m/s - 200 g * Vp
Vb = (250 g * 1.4 m/s - 200 g * Vp) / 50 g
Finally, we need to know the initial velocity of the pendulum (Vp) in order to find the initial velocity of the ball (Vb).