Cheetahs can sustain speeds of 25 m/s over short distances. One of their prey is Thomson's gazelle, with top speed about 22 m/s.
If a 52kg cheetah running southward at top speed grabs and holds onto a 23kg gazelle running westward at 22 m/s, find the velocity of these animals just after the attack. Assume that eastward is a positive x-direction and northward is a positive y-direction.
V^2 = X^2 + Y^2 = 22^2 + 25^2 = 1109
V = 33.3 m/s.
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the attack is equal to the total momentum after the attack.
Let's begin by calculating the initial momentum of the cheetah and the gazelle separately.
The momentum of an object is given by the formula:
Momentum = mass × velocity
For the cheetah:
Mass of the cheetah (m1) = 52 kg
Velocity of the cheetah (v1) = 25 m/s
Momentum of the cheetah (p1) = m1 × v1
For the gazelle:
Mass of the gazelle (m2) = 23 kg
Velocity of the gazelle (v2) = 22 m/s
Momentum of the gazelle (p2) = m2 × v2
Now, let's calculate the initial momentum of both the cheetah and the gazelle:
Initial total momentum (before the attack) = p1 + p2
Next, we need to determine the final velocities of both the cheetah and the gazelle after the attack. Since the cheetah grabs and holds onto the gazelle, they move together as one combined system.
Let's denote the final velocity of the combined system as Vf.
Using the principle of conservation of momentum, the total momentum after the attack is equal to the initial total momentum:
Final total momentum (after the attack) = p1 + p2
Since momentum is a vector quantity, we need to consider the direction of motion as well. The cheetah is running southward (negative y-direction), and the gazelle is running westward (negative x-direction). After the attack, the combined system will have a final velocity, which we need to find.
The momentum in the x-direction (horizontal direction) before and after the attack should be conserved separately. Similarly, the momentum in the y-direction (vertical direction) before and after the attack should be conserved separately.
Applying the principle of conservation of momentum in the x-direction:
Initial momentum in the x-direction = Final momentum in the x-direction
(m1 × v1) + (m2 × v2) = (m1 + m2) × Vfx
Applying the principle of conservation of momentum in the y-direction:
0 = (m1 + m2) × Vfy
Since there is no net force acting in the y-direction (vertical direction) during the attack, the total momentum in that direction remains zero.
We'll solve these equations simultaneously to find the final velocities Vfx and Vfy.
Let's plug in the values:
Cheetah:
m1 = 52 kg
v1 = 25 m/s
Gazelle:
m2 = 23 kg
v2 = 22 m/s
Using the principle of conservation of momentum in the x-direction:
(52 kg × 25 m/s) + (23 kg × -22 m/s) = (52 kg + 23 kg) × Vfx
Using the principle of conservation of momentum in the y-direction:
0 = (52 kg + 23 kg) × Vfy
Solving these equations will give us the final velocities Vfx and Vfy.