Question

An object with total mass m_total = 14.6 kg is sitting at rest when it explodes into three pieces. One piece with mass m₁ = 4.7 kg moves up and to the left at an angle of θ₁ = 20° above the –x axis with a speed of v₁ = 26.8 m/s. A second piece with mass m₂ = 5.1 kg moves down and to the right an angle of θ₂ = 25° to the right of the -y axis at a speed of v₂ = 23.5 m/s.

What is the x-component of the velocity of the third piece?

What is the y-component of the velocity of the third piece?

What is the magnitude of the velocity of the center of mass of the pieces after the collision?

Calculate the increase in kinetic energy of the pieces during the explosion.

Answer 1

m_total = total mass = 14.6 kg

m₁ = 4.7 kg

m₂ = 5.1 kg

m₃ = ?

using the equation

m_total = m₁ + m₂ + m₃

14.6 = 4.7 + 5.1 + m₃

m₃ = 4.8 kg

Consider the motion along the x-direction :

v_1x = v₁ Cos₁ = (26.8) Cos20

v_2x = v₂ Sin₁ = (23.5) Sin25

v_3x = ?

Using conservation of momentum along x-direction

- m₁ v_1x + m₂ v_2x + m₃ v_3x = 0

- (4.7) ((26.8) Cos20) + (5.1) ((23.5) Sin25) + (4.8) v_3x = 0

v_3x = 14.11 m/s

Consider the motion along the y-direction :

v_1y = v₁ Sin₁ = (26.8) Sin20

v_2y = v₂ Cos₁ = (23.5) Cos25

v_3y = ?

Using conservation of momentum along x-direction

m₁ v_1y - m₂ v_2y + m₃ v_3y = 0

(4.7) ((26.8) Sin20) - (5.1) ((23.5) Cos25) + (4.8) v_3y = 0

v_3y = 13.7 m/s

v₃ = sqrt(v_3x² + v_3y²) = sqrt((14.11)² + (13.7)² ) = 19.7 m/s

velocity of center of mass is given as

v_cm = (m₁ v₁ + m₂ v₂ + m₃ v₃) /(m₁ + m₂ + m₃)

v_cm = ((4.7) (26.8) + (5.1) (23.5) + (4.8) (19.7))/(4.7 + 5.1 + 4.8)

v_cm = 23.3 m/s

Increase in kinetic energy is given as

KE = (0.5) (m₁ v²₁ + m₂ v²₂ + m₃ v²₃) = (0.5) ((4.7) (26.8)² + (5.1) (23.5)² + (4.8) (19.7)²) = 4027.5 J