A cat crouches on the floor, at a distance of 1.6 m from a desk chair of height .45 m. The cat jumps onto the chair, landing with zero vertical velocity. The desk chair has frictionless coasters and rolls away when the cat lands. The mass of the cat is 5.7 kg and the mass of the chair is 12 kg.
What is the speed of recoil of the chair and cat?
The time of flight T to the chair can be deduced from the fact that the vertical velocity component is zero at elevation 0.45 m. Thus
(1/2)gT^2 = 0.45 m
T = 0.303 s
Since it travels 1.6 m in that time, you know the horizontal velocity component, 1.6/0.303 = 5.28 m/s
Use that initial (cat) velocity, the chair mass and the law of conservation of momentum to get the speed of recoil of the chair with the cat on it.
5.94m/s
To determine the speed of recoil of the chair and cat, we will use the principle of conservation of momentum.
First, let's calculate the initial momentum of the cat-chair system.
Initial momentum = (Mass of cat * Velocity of cat) + (Mass of chair * Velocity of chair)
We know that the velocity of the cat is zero, and the chair is at rest initially, so the initial momentum is zero.
Next, let's calculate the final momentum of the cat-chair system after the jump.
Final momentum = (Mass of cat * Final Velocity of cat) + (Mass of chair * Final Velocity of chair)
Since the cat lands on the chair with zero vertical velocity, it means the final velocity of the cat and chair will be the same. Let's denote this final velocity as "v".
Final momentum = (Mass of cat * v) + (Mass of chair * v)
Now, according to the principle of conservation of momentum, the initial momentum equals the final momentum:
0 = (Mass of cat * v) + (Mass of chair * v)
Simplifying the equation:
0 = v * (Mass of cat + Mass of chair)
Now we can solve for the final velocity "v".
v = 0 / (Mass of cat + Mass of chair)
v = 0
Therefore, the speed of recoil of the chair and cat is zero. They will come to rest after the jump.
To determine the speed of recoil of the chair and cat, we can use the principle of conservation of momentum. According to this principle, the total momentum before the jump is equal to the total momentum after the jump.
Before the jump, the cat and the chair are stationary, so their initial momentum is zero. After the jump, the cat lands on the chair and both begin to move. Let's assume that the cat and chair move together at a speed of V after the jump.
Now, let's calculate the momentum before and after the jump:
Before the jump:
Since the cat and chair are stationary, the initial momentum is zero.
P_initial = 0
After the jump:
The momentum after the jump can be calculated as the sum of cat's momentum and chair's momentum:
P_final = (mass of cat) × (velocity of cat) + (mass of chair) × (velocity of chair)
In this case, the velocity of the chair is the same as the velocity of the cat after the jump, since they move together.
P_final = (mass of cat) × V + (mass of chair) × V
According to the conservation of momentum, P_initial (zero) is equal to P_final:
0 = (mass of cat) × V + (mass of chair) × V
We can now solve this equation to find the velocity (V) of the cat and chair after the jump. Let's substitute the given values in:
0 = (5.7 kg) × V + (12 kg) × V
0 = 5.7V + 12V
0 = 17.7V
Since the velocity (V) cannot be negative, we take V = 0.
Therefore, the speed of recoil of the chair and cat is 0 m/s.