Question

Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing blow. Asteroid A, which was initially traveling at vA1 = 40.0 m/s with respect to an inertial frame in which asteroid B was at rest, is deflected 30.0 ? from its original direction, while asteroid B travels at 45.0 ? to the original direction of A, as shown in (Figure 1).

a: Find the speed of asteroid A after the collision.

b: Find the speed of asteroid B after the collision.

c: What fraction of the original kinetic energy of asteroid A dissipates during this collision?

Answer 1

We know that, m_A = m_B = m

Given ;   v_A1 = 40 m/s , _A1 = 0 degree , _A2 = 30 degree

_B1 = 0 degree , _B2 = - 45 degree

Using conservation of momentum, we have

On x-axis : m_A v_A1,x + m_B v_B1,x = m_A v_A2,x + m_B v_B2,x

m (40 m/s) + m (0 m/s) = m (v_A2 cos _A2) + m (v_B2 cos _B2)

(40 m/s) = v_A2 cos 30⁰ + v_B2 cos 45⁰

3 v_A2 + 2 v_B2 = (80 m/s)                                                               { eq.1 }

On y-axis :    m_A v_A1,y + m_B v_B1,y = m_A v_A2,y + m_B v_B2,y

m (0 m/s) + m (0 m/s) = m (v_A2 sin _A2) + m (v_B2 sin _B2)

(40 m/s) = v_A2 sin 30⁰ + v_B2 sin 45⁰

v_A2 - 2 v_B2 = (0 m/s)                                                         { eq.2 }

Equating eq.1 & 2, we get

3 v_A2 + 2 v_B2 = (80 m/s)

v_A2 - 2 v_B2 = (0 m/s)   

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(3 + 1) v_A2 + 0 = (80 m/s)

v_A2 = (80 m/s) / (3 + 1)

v_A2 = (80 m/s) / (2.73)

v_A2 = 29.3 m/s

Inserting the value of 'v_A2' in eq.2, we get

v_A2 - 2 v_B2 = (0 m/s)

(29.3 m/s) = 2 v_B2

v_B2 = (29.3 m/s) / (1.41)

v_B2 = 20.7 m/s

(a) Find the speed of asteroid A after the collision.

v_A2 = 29.3 m/s

(b) Find the speed of asteroid B after the collision.

v_B2 = 20.7 m/s

(c) What fraction of the original kinetic energy of asteroid A dissipates during this collision?

An initial kinetic energy will be given as -

K.E_i = (1/2) m_A v_A1² + (1/2) m_B v_B1²

K.E_i = (0.5) m (40 m/s)² + (0.5) m (0 m/s)²

K.E_i = 800 m

A final kinetic energy will be given as -

K.E_f = (1/2) m_A v_A2² + (1/2) m_B v_B2²

K.E_f = (0.5) m (29.3 m/s)² + (0.5) m (20.7 m/s)²

K.E_f = (429.2 m) + (214.2 m)

K.E_f = 643.4 m

Fraction = [(800 m) - (643.4 m)] / (800 m)

Fraction = 0.195