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
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]
