Classical Mechanics

Conservation of Energy

Elastic collisions

         

A body of mass m=5 kgm = 5 \text{ kg} collides elastically with a stationary mass M=9 kgM = 9 \text{ kg} on the xx-axis. Before the collision, the velocity of mass m m is +2 m/s +2 \text{ m/s} . If the velocity of mass MM after the collision is v2=ab m/s, v_2 = \frac{a}{b}\text{ m/s}, where aa and bb are coprime positive integers, what is the value of a+b?a+b?

Two pendulums with respective masses of m1=5 kg m_1 = 5 \text{ kg} and m2=10 kg m_2 = 10 \text{ kg} hang from the same point by identical strings, as shown in the figure above. Both being initially h=1 m h = 1 \text{ m} high from the lowest point, they swing down and collide elastically at the lowest point. If the velocity of mass m1 m_1 after the collision v1v_1 satisfies v12=ab, v_1^2=\frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+b?a+b?

A body of mass m1=5 kg m_1 = 5 \text{ kg} collides elastically with a stationary mass m2=4 kg. m_2 = 4 \text{ kg}. Before the collision, the velocity of mass m1 m_1 is u=2m/s. u = 2 \text{m/s} . After collision, masses m1 m_1 and m2 m_2 move in directions that make respective angles of θ1\theta_1 and θ2 \theta_2 with the original direction that m1m_1 had moved in. If sinθ1=45 \sin{\theta_1} = \frac{4}{5} , what is the velocity of mass m1 m_1 after the collision?

Consider a one dimensional collision between two point masses with different masses m1m_1 and m2m_2 and initial velocities v1v_1 and v2v_2. We define the ratio rr as the ratio of the difference in initial velocities to the difference in final velocities:

r=v1,iv2,iv1,fv2,fr=\frac{|v_{1,i}-v_{2,i}|} {|v_{1,f}-v_{2,f}|}.

What is rr if the collision is perfectly elastic?

Two pendulums with respective masses of m1=4 kg m_1 = 4 \text{ kg} and m2=8 kg m_2 = 8 \text{ kg} hang from the same point by identical strings, as shown in the figure above. Both being initially h=18 m h = 18 \text{ m} high from the lowest point, they swing down and collide elastically at the lowest point. What is the maximum height that pendulum m1m_1 reaches after the collision?

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