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A few Numerical Problems from Mechanics

ninzyy posted:

1. A 12 kg block sliding at 6.5 m/s along a level floor encounters rough area where the coefficient of friction is 0.20. The rough area is 5 meters long. How fast will the block be moving as it leaves the rough area? Where did the energy go?

2. A 800 kg car traveling southward at 9.2 m/s collides with a 900 kg car traveling northward at 4.5 m/s.?
The cars become entangled. What is the speed and direction of the cars immediately following the collision?

3. A 6kg block slides down a 2.0 m long frictionless ramp which is inclined 40 degrees above the ground.?
a) find the height of the top of the ramp and then determine the speed at the bottom of the ramp.

b) If the block collides with, and sticks to a 8.2 kg block which had been at rest at the bottom of the ramp, how fast would they now move?

c) Determine how much energy the two blocks have after the collision.

1. The problem can be solved using work energy theorem

i.e; work done against friction = change in KE
$\mu mg x = \frac{1}{2}m\left (v_{1}^{2}-v_{2}^{2} \right ) \Rightarrow v_{1}^{2}-v_{2}^{2}= 2\mu g x$
Substitute the values and you will get the answer.

2. This is an example for inelastic collision.

According to the law of conservation of momentum,

total momentum before collision = total momentum after collision

Taking northwards as positive and southwards as negative,

Substitute in the formula
$m_{1}v_{1}+m_{2}v_{2}=\left ( m_{1}+m_{2}\right )v$

Substitute the values of m1, m2, v1 and v2 and you can calculate the value of v.

If v is positive, it will be towards north and if negative it will be towards south.

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