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_1 ) immediately before the collision?
( v_1=) 7.28 m/s
what was nates speed ( v_1 ) immediately after the collision?
( v_1=) - 5.45 m/s

Explanation:

Step1: Apply conservation of momentum

$$m_1u_1 + m_2u_2=m_1v_1 + m_2v_2$$
Here, \(m_1 = 129\space kg\), \(m_2=897\space kg\), \(u_2 = 0\space m/s\), \(v_2=1.83\space m/s\)
So, \(129u_1+897\times0 = 129v_1+897\times1.83\)
\(129u_1=129v_1 + 1641.51\)

Step2: Apply coefficient of restitution (\(e = 1\) for elastic collision)

$$e=\frac{v_2 - v_1}{u_1 - u_2}$$
Since \(e = 1\) and \(u_2 = 0\)
\(1=\frac{1.83 - v_1}{u_1-0}\)
\(u_1=1.83 - v_1\)

Step3: Substitute \(u_1\) in momentum equation

Substitute \(u_1 = 1.83 - v_1\) into \(129u_1=129v_1 + 1641.51\)
\(129(1.83 - v_1)=129v_1 + 1641.51\)
\(236.07-129v_1=129v_1 + 1641.51\)
\(-129v_1-129v_1=1641.51 - 236.07\)
\(-258v_1=1405.44\)
\(v_1=- 5.45\space m/s\)

Step4: Find \(u_1\)

Substitute \(v_1=-5.45\) into \(u_1 = 1.83 - v_1\)
\(u_1=1.83-(-5.45)=7.28\space m/s\)

Answer:

Nate's speed immediately before the collision \(u_1 = 7.28\space m/s\) and immediately after the collision \(v_1=-5.45\space m/s\)