Question

example:
a ball of putty with a mass of 0.250 kg is thrown with a velocity of 7.50 m/s at a 0.500 kg
ball resting nearby. the putty sticks to the ball. what is the velocity of the putty - ball system
after the collision?
____, __, __, ____ kg·m/s.

Explanation:

Step1: Identify the principle (Conservation of Momentum)

The collision is inelastic (putty sticks to the ball), so we use the law of conservation of momentum, which states that the total initial momentum equals the total final momentum. The formula is \( m_1v_1 + m_2v_2=(m_1 + m_2)v_f \), where \( m_1 = 0.250\space kg \), \( v_1 = 7.50\space m/s \), \( m_2 = 0.500\space kg \), \( v_2 = 0\space m/s \) (since the ball is at rest), and \( v_f \) is the final velocity of the combined system.

Step2: Calculate initial momentum

First, calculate the initial momentum of the putty: \( p_1 = m_1v_1=0.250\space kg\times7.50\space m/s = 1.875\space kg\cdot m/s \). The initial momentum of the ball is \( p_2 = m_2v_2 = 0.500\space kg\times0\space m/s = 0\space kg\cdot m/s \). So the total initial momentum \( p_{total\space initial}=p_1 + p_2 = 1.875\space kg\cdot m/s+ 0\space kg\cdot m/s = 1.875\space kg\cdot m/s \).

Step3: Calculate final mass

The final mass of the combined system is \( m_{total}=m_1 + m_2 = 0.250\space kg+ 0.500\space kg = 0.750\space kg \).

Step4: Solve for final velocity

Using the conservation of momentum formula \( p_{total\space initial}=m_{total}v_f \), we can solve for \( v_f \): \( v_f=\frac{p_{total\space initial}}{m_{total}}=\frac{1.875\space kg\cdot m/s}{0.750\space kg}=2.50\space m/s \). But the question seems to ask for momentum? Wait, no, the units given are \( kg\cdot m/s \), but momentum of the combined system after collision is the same as total initial momentum, which is \( 1.875\space kg\cdot m/s \)? Wait, no, maybe there's a misinterpretation. Wait, the problem statement's blank is confusing, but if we follow the conservation of momentum, the total momentum before and after is the same. Wait, maybe the question is to fill in the values for the calculation steps? Wait, the initial momentum of putty is \( 0.250\times7.50 = 1.875 \), initial momentum of ball is \( 0.500\times0 = 0 \), total initial momentum is \( 1.875+0 = 1.875 \), and the final momentum is also \( 1.875 \) (since momentum is conserved). But the units are \( kg\cdot m/s \). So the blanks could be filled with the steps: first blank: \( 0.250\times7.50 = 1.875 \), second blank: \( 0.500\times0 = 0 \), third blank: \( 1.875 + 0 = 1.875 \), fourth blank: \( 1.875 \) (since final momentum equals initial momentum). Wait, but maybe the question is to find the final velocity, but the units are \( kg\cdot m/s \), which is momentum. So the total momentum after collision is equal to total initial momentum, which is \( 1.875\space kg\cdot m/s \). But the blanks are four, so maybe:

First blank: \( 0.250\times7.50 = 1.875 \)

Second blank: \( 0.500\times0 = 0 \)

Third blank: \( 1.875+0 = 1.875 \)

Fourth blank: \( 1.875 \) (since final momentum is same as initial)

Wait, but maybe the problem has a typo, but following the steps:

Answer:

\( 1.875 \), \( 0 \), \( 1.875 \), \( 1.875 \)