Completely lost on what formula to use: A 50 gram ball enters a pendulum with mass 200 g. The pair then swings up to a height of 12 cm. Find the velocity at which the pair move immediately after the collision. Find the initial velocity of the ball before the collision. Appreciate anyone's guidance.
ok, the masses moved up 12 cm, so the change in Potential Energy is mgh=(.250*9.8*.12)
Now that energy must have came from initial Kinetic energy, so
1/2 m vi^2=mgh
vi=sqrt(2*9.8*.12) That is the velocity of the pair after collision.
Now, at collision, conservation of momentum applies
mb*vb=(mb+mp)Vi
.05*Vball=(.250)sqrt*2*9.8*.12)
solve for vball.
Works perfectly. Thanks so much!
To solve this problem, we can use the principles of conservation of momentum and conservation of mechanical energy.
Let's start by finding the velocity at which the pair moves immediately after the collision:
1. Conservation of momentum:
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. In this case, the initial momentum is equal to the final momentum.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v):
p = m * v
Before the collision:
The momentum of the ball before the collision is given by:
p_ball_initial = m_ball * v_ball_initial
The momentum of the pendulum before the collision is given by:
p_pendulum_initial = m_pendulum * v_pendulum_initial
After the collision:
The momentum of the ball after the collision is given by:
p_ball_final = m_ball * v_ball_final
The momentum of the pendulum after the collision is given by:
p_pendulum_final = m_pendulum * v_pendulum_final
Since the pair is swinging up, the pendulum momentarily stops after the collision. Therefore, the momentum of the pendulum after the collision is zero:
p_pendulum_final = 0
Since the initial momentum is equal to the final momentum, we have:
p_ball_initial + p_pendulum_initial = p_ball_final + p_pendulum_final
Therefore, we can rewrite this equation as:
m_ball * v_ball_initial + m_pendulum * v_pendulum_initial = m_ball * v_ball_final + 0
Now we can solve this equation to find the value of v_ball_final.
2. Conservation of mechanical energy:
According to the conservation of mechanical energy, the total mechanical energy before the collision is equal to the total mechanical energy after the collision.
The mechanical energy (E) of an object is given by the sum of its kinetic energy (KE) and its potential energy (PE):
E = KE + PE
Before the collision:
The mechanical energy of the ball before the collision is given by:
E_ball_initial = KE_ball_initial + PE_ball_initial
But the ball is at the same height before and after the collision, so its potential energy is the same:
PE_ball_initial = PE_ball_final
After the collision:
The mechanical energy of the ball after the collision is given by:
E_ball_final = KE_ball_final + PE_ball_final
Since the initial mechanical energy is equal to the final mechanical energy, we have:
E_ball_initial = E_ball_final
Substituting the expressions for the mechanical energy, we have:
KE_ball_initial + PE_ball_initial = KE_ball_final + PE_ball_final
Now we can proceed to find the initial velocity of the ball (v_ball_initial).
To summarize:
- Use the conservation of momentum equation (m_ball * v_ball_initial + m_pendulum * v_pendulum_initial = m_ball * v_ball_final) to find the velocity of the ball after the collision (v_ball_final).
- Use the conservation of mechanical energy equation (KE_ball_initial + PE_ball_initial = KE_ball_final + PE_ball_final) to find the initial velocity of the ball (v_ball_initial).
Remember to use the given values in the problem (mass of the ball, mass of the pendulum, and the height).