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A hungry 173 kg lion running northward at 82.7 km/hr attacks and holds onto a 36.2 kg Thomson's gazelle running eastward at 57.4 km/hr. Find the final speed of the lion–gazelle system immediately after the attack.

Answer :

Answer:

the final speed after attack is 69.09 km/h

Explanation:

lion weight = 173 kg

running northward at 82.7 km/hr

weight of gazelle = 36.2 kg

running eastward at 57.4 km/hr

using momentum conservation along north direction

(173) × (82.7) = (173 + 36.2 ) × vₙ

vₙ = 68.38 km/h

using momentum conservation along east direction

36.2 × 57.4 = (173 + 36.2 ) × vₓ

vₓ = 9.93 km/h

[tex]v = \sqrt{v^2_n+v^2_x} \\v = \sqrt{68.38^2+9.93^2}\\v=69.09 km/h[/tex]

hence the final speed after attack is 69.09 km/h

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Rewritten by : Barada

The final speed of the lion-gazelle system immediately after the attack is approximately 19.22 m/s.

Calculating the Final Speed of the Lion-Gazelle System

To determine the final speed of the combined lion-gazelle system after the attack, we will use the principle of conservation of momentum. Since the momentum is a vector quantity, we need to consider both the northward and eastward directions independently.

Step 1: Determine Individual Momentums

  • Momentum of the lion: [tex]p_{lion} = mass_{lion} \times velocity_{lion}[/tex]
  • Momentum of the gazelle: [tex]p_{gazelle} = mass_{gazelle} \times velocity_{gazelle}[/tex]

Given data:

  • [tex]mass_{lion} = 173 kg[/tex]
  • [tex]velocity_{lion} = 82.7 km/hr[/tex] (converted to m/s: [tex]82.7 \times 1000 / 3600 = 22.97 m/s[/tex])
  • [tex]mass_{gazelle} = 36.2 kg[/tex]
  • [tex]velocity_{gazelle} = 57.4 km/hr[/tex] (converted to m/s: [tex]57.4 \times 1000 / 3600 = 15.94 m/s[/tex])

Calculate individual momentums in x (eastward for gazelle) and y (northward for lion) directions:

  • [tex]p_{lion} = 173 kg \times 22.97 m/s = 3973.81 kg.m/s (northward)[/tex]
  • [tex]p_{gazelle} = 36.2 kg \times 15.94 m/s = 577.60 kg.m/s (eastward)[/tex]

Step 2: Calculate Combined Momentum and Final Speed

The system's total momentum[tex](P_{total})[/tex] is the vector sum of the individual momentums:

  • [tex]P_{total} = \sqrt{(p_{lion}^2 + p_{gazelle}^2)}[/tex]
  • [tex]P_{total} = \sqrt{(3973.81^2 + 577.60^2)} = 4020.21 kg.m/s[/tex]

Combine the masses:

  • [tex]mass_{total} = mass_{lion} + mass_{gazelle}[/tex]
  • [tex]mass_{total} = 173 kg + 36.2 kg = 209.2 kg[/tex]

Finally, the velocity (v_final) of the combined system is:

  • [tex]v_{final} = P_{total} / mass_{total}[/tex]
  • [tex]v_{final} = 4020.21 kg.m/s / 209.2 kg = 19.22 m/s[/tex]