A 20.0 kg wood ball hangs from a 2.10 m -long wire. The maximum tension the wire can withstand without breaking is 500 N . A 0.900 kg projectile traveling horizontally hits and embeds itself in the wood ball.
To find the impact force exerted by the projectile on the wood ball, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
The momentum of an object can be calculated by multiplying its mass by its velocity. Since the projectile is traveling horizontally, its velocity can be considered as the horizontal component only.
Before the collision:
Momentum of the projectile = mass of the projectile * velocity of the projectile
Momentum of the wood ball = mass of the wood ball * velocity of the wood ball (initially at rest)
After the collision:
Total momentum = (mass of the projectile + mass of the wood ball) * final velocity
Since the projectile embeds itself in the wood ball, the final velocity would be the same for both the ball and the projectile.
Now we can set up the equation for conservation of momentum:
(mass of the projectile * velocity of the projectile) + (mass of the wood ball * velocity of the wood ball) = (mass of the projectile + mass of the wood ball) * final velocity
To solve for the final velocity, we can rearrange the equation as follows:
final velocity = [(mass of the projectile * velocity of the projectile) + (mass of the wood ball * velocity of the wood ball)] / (mass of the projectile + mass of the wood ball)
Given:
mass of the projectile (m_proj) = 0.900 kg
velocity of the projectile (v_proj) = ?
mass of the wood ball (m_wood) = 20.0 kg
velocity of the wood ball (v_wood) = 0 m/s (initially at rest)
Using the conservation of momentum equation:
final velocity = [(0.900 kg * v_proj) + (20.0 kg * 0 m/s)] / (0.900 kg + 20.0 kg)
Simplifying the equation:
final velocity = (0.900 kg * v_proj) / 20.9 kg
Now, to find the impact force exerted by the projectile on the wood ball, we can use Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of momentum.
Force = mass * acceleration
Since momentum is mass times velocity, the rate of change of momentum is equal to the mass times acceleration. In this case, the final velocity is the change in velocity (from initially at rest to the final velocity).
Force = (mass of the wood ball + mass of the projectile) * (final velocity - initial velocity)
Given:
mass of the wood ball (m_wood) = 20.0 kg
initial velocity of the wood ball (v_wood_initial) = 0 m/s (initially at rest)
final velocity of the wood ball (v_wood_final) = final velocity (from the conservation of momentum equation)
force (F) = ?
Using Newton's second law of motion:
Force = (20.0 kg + 0.900 kg) * (final velocity - 0 m/s)
Simplifying the equation:
Force = 20.9 kg * final velocity
Now we have the equation to calculate the impact force exerted by the projectile on the wood ball. We can substitute the final velocity obtained from the conservation of momentum equation to find the final answer.