0.6 6.7 collisions of point masses in two dimensions  (Page 4/5)

Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of $\text{30}\text{.}\mathrm{0º}$ ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that ${\theta }_1-{\theta }_2=\text{90º}$ for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.

Confirm that the results of the example [link] do conserve momentum in both the $x$ - and $y$ -directions.

A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of $\text{20}\text{.}\mathrm{0º}$ above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired?

Professional Application

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei $\left({}^4\text{He}\right)$ from gold-197 nuclei $\left({}^{\text{197}}\text{Au}\right)$ . The energy of the incoming helium nucleus was $8.00\times {\text{10}}^{-\text{13}}\text{J}$ , and the masses of the helium and gold nuclei were $6.68\times {\text{10}}^{-\text{27}}\text{kg}$ and $3.29\times {\text{10}}^{-\text{25}}\text{kg}$ , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of $\text{120º}$ during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?

Starting with equations $m_1{v}_1=m_1{v\prime }_1^{}\text{cos}{\theta }_1+m_2{v\prime }_2^{}\text{cos}{\theta }_2$ and $0=m_1{v\prime }_1^{}\text{sin}{\theta }_1+m_2{v\prime }_2^{}\text{sin}{\theta }_2$ for conservation of momentum in the $x$ - and $y$ -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,

as discussed in the text.

Integrated Concepts

A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?