Question

A projectile of mass m1 moving with speed v1 in the +x direction strikes a stationary target of mass m2 head-on. The collision is elastic. Use the Momentum Principle and the Energy Principle to determine the final velocities of the projectile and target, making no approximations concerning the masses. (Use the following as necessary: m1, m2, and v1.)

Answer 1

Answer:

$v''=\frac{2m_{1}v_{1}}{m_{1}+m_{2}}$

$v' = \frac{m_{1}-m_{2}}{m_{1}+m_{2}}v_{1}$

Explanation:

first mass, m1

second mass, m2

initial velocity of m1 is v1.

initial velocity of m2 is zero.

Let the final velocity of m1 is v' and the final velocity of m2 is v''.

Collision is elastic so the coefficient of restitution is 1.

By using the conservation of momentum

m1 x v1 + m2 x 0 = m1 x v + m2 x v''

m1 v1 = m1 v' + m2 v'' .... (1)

by the formula of coefficient of restitution

$1=\frac{v''-v'}{v_{1}-0}$

v1 = v'' - v' .... (2)

So, v' = v'' - v1 put in equation (1)

m1 v1 = m1 x (v'' - v1) + m2 v''

m1 v1 = m1 v'' - m1 v1 + m2 v''

2 m1 v1 = (m1 + m2) v''

$v''=\frac{2m_{1}v_{1}}{m_{1}+m_{2}}$

so,

$v' = \frac{2m_{1}v_{1}}{m_{1}+m_{2}}-v_{1}$

$v' = \frac{m_{1}-m_{2}}{m_{1}+m_{2}}v_{1}$

Answer 2

Using Momentum Principle and the Energy Principle to determine the final velocities of the projectile and target are respectively;

v1 = (m1•u1 - m2•u1)/(m1 + m2)

v1 = (m1•u1 - m2•u1)/(m1 + m2)v2 = (m1•u1 + m1•u2)/(m1 + m2)

Let us denote the following;

Mass of projectile; m1

Mass of target; m2

Initial velocity of projectile; u1

Final velocity of projectile; v1

Initial velocity of target; u2 = 0 m/s

Final velocity of target; v2

From conservation of momentum, we have;

(m1•u1) + (m2•u2) = (m1•v1) + (m2•v2)

Plugging in the relevant values;

(m1•u1) + (m2•0) = (m1•v1) + (m2•v2)

(m1•u1) = (m1•v1) + (m2•v2) - - - (eq 1)

From energy principle, we have;

u1 + v1 = u2 + v2

Thus;

Since u2 is zero, then we have;

u1 + v1 = v2 - - - (eq 2)

Put (u1 + v1) for v2 in eq 1 to get;

(m1•u1) = (m1•v1) + (m2(u1 + v1))

(m1•u1) = (m1•v1) + (m2•u1 + m2•v1) - - - (eq 3)

Let us make v1 the subject;

m1•u1 - m2•u1 = m1•v1 + m2•v1

v1(m1 + m2) = m1•u1 - m2•u1

v1 = (m1•u1 - m2•u1)/(m1 + m2)

Let us make v1 the subject in eq 2;

v1 = v2 - u2

Put v2 - u2 for v1 in eq 1 to get;

(m1•u1) = (m1(v2 - u2) + (m2•v2)

(m1•u1) = m1•v2 - m1•u2 + m2•v2

m1•u1 + m1•u2 = v2(m1 + m2)

v2 = (m1•u1 + m1•u2)/(m1 + m2)

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