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Anna11 [10]
2 years ago
7

A bullet of mass M1 is fired towards a block of mass m2 initially at rest at the edge of a frictionless table of height h as in

the figure. The initial speed of the bullet is vi . Consider two cases, a completely inelastic one and an elastic one,where the bullet bounces off the block. inelastic case elastic case a bullet inside no bullet inside A B A' B' What is the ratio of the flight time; i.e., tAB tA′B′ ?
Physics
1 answer:
pantera1 [17]2 years ago
4 0

The ratio of time of flight for inelastic collision to elastic collision is 1:2

The given parameters;

  • <em>mass of the bullet, = m₁</em>
  • <em>mass of the block, = m₂</em>
  • <em>initial velocity of the bullet, = u₁</em>
  • <em>initial velocity of the block, = u₂</em>

Considering inelastic collision, the final velocity of the system is calculated as;

m_1u_1 + m_2u_2 = v(m_1 + m_2)\\\\m_1u_1 + 0 = v(m_1 + m_2)\\\\v= \frac{m_1u_1}{m_1 + m_2} \ -- (1)\\\\

The time of motion of the system form top of the table is calculated as;

v = u + gt\\\\v = 0 + gt\\\\v = gt\\\\t= \frac{v}{g} \\\\t_A = \frac{m_1u_1}{g(m_1 + m_2)} \ \ ---(2)

Considering elastic collision, the final velocity of the system is calculated as;

m_1u_1 + m_2 u_2 = m_1v_1 + m_2v_2\\\\m_1u_1 + 0 = m_1v_1 + m_2v_2\\\\m_1 u_1 = m_1v_1 + m_2v_2

Apply one-directional velocity

u_1 + (-v_1) = u_2 + v_2\\\\u_1 -v_1 = 0 + v_2\\\\v_1 = v_2 -u_1

Substitute the value of v_1 into the above equation;

m_1u_1 = m_1(v_2 - u_1) + m_2 v_2\\\\m_1u_1 = m_1v_2 -m_1u_1 + m_2v_2\\\\2m_1u_1 = m_1v_2 + m_2v_2\\\\2m_1u_1= v_2(m_1 + m_2)\\\\v_2 = \frac{2m_1u_1}{m_1+ m_2}  \ --(3)

where;

v_2 is the final velocity of the block after collision

<em>Since the</em><em> bullet bounces off</em><em>, we assume that </em><em>only the block fell </em><em>to the ground from the table.</em>

The time of motion of the block is calculated as follows;

v_2 = v_0 + gt\\\\v_2 = 0 + gt\\\\t = \frac{v_2}{g} \\\\t_B = \frac{v_2}{g} \\\\ t_B = \frac{2m_1u_1}{g(m_1 + m_2)} \ \ ---(4)

The ratio of time of flight for inelastic collision to elastic collision is calculated as follows;

\frac{t_A}{t_B} = \frac{m_1u_1}{g(m_1 + m_2)} \times \frac{g(m_1 + m_2)}{2m_1u_1} \\\\\frac{t_A}{t_B} = \frac{1}{2} \\\\t_A:t_B = 1: 2

Learn more about elastic and inelastic collision here: brainly.com/question/7694106

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A van starts off 152 miles directly north from the city of Springfield. It travels due east at a speed of 25 miles per hour. Aft
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Answer:

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Explanation:

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Vertical distance of starting point of van from Springfield (d) = 152 miles

Speed in east direction (s) = 25 mph

Distance traveled in east direction (e) = 91 miles

Let the direct distance from Springfield of the van be 'x' at any time 't'.

Now, from the question, it is clear that, the vertical distance of van is fixed at 152 miles and only the horizontal distance is changing with time.

Now, consider a right angled triangle SNE representing the given situation.

Point S represents Springfield, N represents the starting point of van and E represents the position of van at any time 't'.

SN = d = 152 miles (fixed)

Now, using the pythagorean theorem, we have:

SE^2=SN^2+NE^2\\\\x^2=d^2+e^2\\\\x^2=(152)^2+e^2----(1)

Now, differentiating both sides with respect to time 't', we get:

2x\frac{dx}{dt}=0+2e\frac{de}{dt}\\\\\frac{dx}{dt}=\frac{e}{x}\frac{de}{dt}

Now, we are given speed as 25 mph. So, \frac{de}{dt}=25\ mph

Also, when e=91\ mi, we can find 'x' using equation (1). This gives,

x^2=23104+(91)^2\\\\x=\sqrt{31385}=177.16\ mi

Now, plug in the values of 'e' and 'x' and solve for \frac{dx}{dt}. This gives,

\frac{dx}{dt}=\frac{91}{177.16}\times 25\\\\\frac{dx}{dt}=12.84\ mph

Therefore, the distance between the van and Springfield is changing at a rate of 12.84 miles per hour

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