A 5000 kg cart carrying a vertical rocket launcher moves to the right at a constant speed of 27.0 m/s along a horizontal track. It launches a 54.8 kg- rocket vertically upward with an initial speed of 33.1 m/s relative to the cart.
A) where will the rocket land, relatively to the cart?
B)How far does the cart move while the rocket is in the air?
A) it lands in the cart. think about why.
B) multiply the time the rocket is in the air by 27 m/s.
A real rocket accelerates while burning. This a rather phony problem
To answer these questions, we can use the conservation of momentum. The total momentum before the launch should be equal to the total momentum after the launch.
Let's calculate the initial momentum of the cart:
Momentum of the cart = mass of the cart * velocity of the cart
Momentum of the cart = 5000 kg * 27.0 m/s
Momentum of the cart = 135,000 kg * m/s
Next, let's calculate the initial momentum of the rocket:
Momentum of the rocket = mass of the rocket * velocity of the rocket
Momentum of the rocket = 54.8 kg * 33.1 m/s
Momentum of the rocket = 1,811.88 kg * m/s
Now, let's calculate the total momentum after the launch. The rocket and the cart will move in opposite directions, so we need to consider the signs:
Total momentum after the launch = momentum of the cart - momentum of the rocket
Total momentum after the launch = 135,000 kg * m/s - 1,811.88 kg * m/s
Total momentum after the launch = 133,188.12 kg * m/s
A) To determine where the rocket will land relative to the cart, we need to consider that the total momentum after the launch is nonzero. This means that there is an external force acting on the system of the cart and the rocket.
Since the total momentum after the launch is nonzero and the rocket is launched vertically upward, it indicates that there is a net force acting on the rocket (upward) and on the cart (downward). This force is not balanced, so the cart and the rocket will experience an acceleration in the opposite directions.
Therefore, the rocket will land in a direction opposite to its initial motion, relative to the cart. In this case, the rocket will land behind the cart.
B) To determine how far the cart moves while the rocket is in the air, we need to calculate the time it takes for the rocket to land and then multiply it by the velocity of the cart.
First, let's calculate the time it takes for the rocket to land:
Vertical distance covered by the rocket = (initial vertical velocity of the rocket)^2 / (2 * gravitational acceleration)
Vertical distance covered by the rocket = (33.1 m/s)^2 / (2 * 9.8 m/s^2)
Vertical distance covered by the rocket = 554.41 m
Time taken by the rocket to land = (2 * vertical distance covered by the rocket) / (initial vertical velocity of the rocket)
Time taken by the rocket to land = (2 * 554.41 m) / (33.1 m/s)
Time taken by the rocket to land = 33.51 s
Now, let's calculate the distance covered by the cart while the rocket is in the air:
Distance covered by the cart = velocity of the cart * time taken by the rocket to land
Distance covered by the cart = 27.0 m/s * 33.51 s
Distance covered by the cart = 905.77 m
Therefore, the cart moves approximately 905.77 meters while the rocket is in the air.
To answer these questions, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, provided there are no external forces acting on the system.
A) To determine where the rocket will land relative to the cart, we need to calculate the horizontal distance traveled by the cart while the rocket is in the air. Since there are no external forces acting horizontally on the system, the momentum of the cart-rocket system in the horizontal direction remains constant.
The initial momentum of the cart-rocket system in the horizontal direction is given by:
Initial momentum = mass of the cart * velocity of the cart
Initial momentum = (5000 kg) * (27.0 m/s) = 135,000 kg*m/s
The final momentum of the cart-rocket system in the horizontal direction remains constant, as there are no external forces acting horizontally. Therefore, the final momentum is also 135,000 kg*m/s.
Since the mass of the rocket is much smaller than the mass of the cart, the momentum of the rocket can be neglected for this calculation.
Now, let's assume that the cart moves a horizontal distance of 'x' meters while the rocket is in the air. The final velocity of the cart is then given by:
Final velocity = final momentum / mass of the cart
Since the final momentum is equal to the initial momentum, we have:
Final velocity = (135,000 kg*m/s) / (5000 kg) = 27.0 m/s
Since the cart is moving at a constant speed of 27.0 m/s, we can conclude that it does not change its position horizontally while the rocket is in the air. Therefore, the rocket will land at the same horizontal position as where it was launched from relative to the cart.
B) As mentioned before, the cart does not change its position horizontally while the rocket is in the air. Therefore, the distance traveled by the cart while the rocket is in the air is zero.