Question

nate the skate was an avid physics student whose main non - physics interest in life was high - speed skateboarding. in particular,
nate would often don a protective suit of bounce - tex, which he invented. then, after working up a high speed on his
skateboard, would collide with some object. in this way, he got a gut feel for the physical properties of collisions.
on one occasion, nate, whose mass (including his armor) was 129 kg, hurled himself against a 897 kg stationary statue of isaac
newton in a perfectly elastic linear collision. as a result, isaac started moving at 1.83 m/s and nate bounced backward.
ignore friction with the ground.
what was nates speed ( v_i ) immediately before the collision?
( v_i=)7.28 m/s
correct answer
what was nates speed ( v_f ) immediately after the collision?
( v_f=) - 5.45 m/s
incorrect answer

Explanation:

Step1: Conservation of momentum and kinetic energy in elastic collision

For a perfectly elastic linear collision, we use the conservation of momentum \(m_1v_{i1}+m_2v_{i2}=m_1v_{f1}+m_2v_{f2}\) and conservation of kinetic energy \(\frac{1}{2}m_1v_{i1}^2+\frac{1}{2}m_2v_{i2}^2=\frac{1}{2}m_1v_{f1}^2+\frac{1}{2}m_2v_{f2}^2\). Given \(m_1 = 129\space kg\), \(v_{i2}=0\space m/s\), \(m_2 = 897\space kg\), \(v_{f2}=1.83\space m/s\).
From conservation of momentum: \(m_1v_{i1}=m_1v_{f1}+m_2v_{f2}\). From conservation of kinetic energy: \(v_{i1}-v_{i2}=v_{f2}-v_{f1}\), since \(v_{i2} = 0\), we have \(v_{i1}=v_{f2}-v_{f1}\), so \(v_{f1}=v_{f2}-v_{i1}\).
Substitute \(v_{f1}\) into the momentum equation: \(m_1v_{i1}=m_1(v_{f2}-v_{i1})+m_2v_{f2}\).
Expand: \(m_1v_{i1}=m_1v_{f2}-m_1v_{i1}+m_2v_{f2}\).
Rearrange: \(2m_1v_{i1}=(m_1 + m_2)v_{f2}\).
Solve for \(v_{i1}\): \(v_{i1}=\frac{(m_1 + m_2)v_{f2}}{2m_1}=\frac{(129 + 897)\times1.83}{2\times129}=\frac{1026\times1.83}{258}=7.28\space m/s\).

Step2: Find \(v_{f}\)

From \(v_{i1}=v_{f2}-v_{f1}\), so \(v_{f1}=v_{f2}-v_{i1}\). Substitute \(v_{i1} = 7.28\space m/s\), \(v_{f2}=1.83\space m/s\). \(v_{f1}=1.83 - 7.28=- 5.45\space m/s\).

Answer:

\(v_{i}=7.28\space m/s\), \(v_{f}=-5.45\space m/s\)