A cart running on frictionless air tracks is propelled by a stream of water expelled by a gas-powered pressure washer stationed on the cart. There is a 1.0-m3 water tank on the cart to provide the water for the pressure washer. The mass of the cart, including the operator riding it, the pressure washer with its fuel, and the empty water tank, is 430 kg. The water can be directed, by switching a valve, either backward or forward. In both directions, the pressure washer ejects 210 L of water per min with a muzzle velocity of 26.0 m/s.
(a) If the cart starts from rest, after what time should the valve be switched from backward (forward thrust) to forward (backward thrust) for the cart to end up at rest?
(b) What is the mass of the cart at that time, and what is its velocity? (Hint: It is safe to neglect the decrease in mass due to the gas consumption of the gas-powered pressure washer!)
mass
velocity
magnitude
direction
(c) What is the magnitude of the thrust of this "rocket"?
(d) What is the acceleration of the cart immediately before the valve is switched?
magnitude
direction
The mass decreases while the water jet is blasting, so the acceleration rate constantly increases. Then the thrust direction changes, but the magnitude of the acceleration continues to increase.
The trick to this problem is not to try to answer the questions in the order they are asked.
Let's do (c) first. It's easier that way. The thrust is
|Vexhaust*(dm/dt)|
= 26.0 m/s*(210 kg/60s) = 91 Newtons
The initial mass of the cart, with water, is 1430 kg. When all the water is gone, the final mass is 430 kg.
The velocity change during a period the jet is aimed in one direction is
deltaV = Vexhaust*ln (M1/M2).
Here's why:
Thrust = Vexhaust*dM/dt = M dV/dt
dV = Vexhaust*dM/M
V2 - V1 = Vexhaust*ln(M1/M2)
If you want the velocity change after thrust reversal to equal (but be in the opposite direction of) that before reversal, the intermediate mass M2 must satisfy
M1/M2 = M2/M3
M2 = sqrt(M1*M3) = sqrt(1430*430) = 784 kg. That answers part of (b).
For the velocity then, it is
Vexhaust*lm(M1/M2) = 26.0 ln(1430/784) = 15.6 m/s. So much for part (b)
Since you started with a mass of 1430 kg and dropped to 784 kg, the time to reverse thrust is after you have lost 646 kg at 3.5 kg/s, which means after 185 seconds. So much for part a.
For (d), use a = F/m, with M = 784 kg and F = 91 N
a = 0.116 m/s^2
To solve this problem, we need to apply the principle of conservation of momentum.
Let's begin by finding the initial momentum of the cart (including the operator, pressure washer, fuel, and empty water tank) before the valve is switched. Since the cart starts from rest, the initial velocity is 0.
(a) The momentum can be calculated using the formula:
initial momentum = mass of the cart * initial velocity
Given:
Mass of the cart (m) = 430 kg
Initial velocity (v) = 0 m/s
The initial momentum of the cart before the valve is switched is:
initial momentum = 430 kg * 0 m/s = 0 kg*m/s
Now, let's determine the momentum of the cart after the valve is switched to forward (backward thrust) using the given information.
(b) The mass of the cart at that time remains the same at 430 kg, as it explicitly states that the decrease in mass due to gas consumption can be neglected. The velocity of the cart after the valve is switched will depend on the time it takes to switch the valve.
The change in momentum can be calculated using the formula:
change in momentum = mass of the cart * final velocity - initial momentum
Since we want the cart to come to rest, the final velocity would be 0 m/s.
Therefore, the change in momentum would be:
change in momentum = 430 kg * 0 m/s - 0 kg*m/s = 0 kg*m/s
To make the change in momentum equal to zero, the valve should be switched at a time when the cart has moved in the forward direction and gained momentum equal to its initial momentum.
(c) The magnitude of the thrust of this "rocket" can be calculated as the rate of change of momentum. Since the change in momentum is zero, the magnitude of the thrust is also zero. This means that the pressure washer does not provide any net thrust.
(d) The acceleration of the cart immediately before the valve is switched can be calculated using the formula:
acceleration = change in velocity / time taken
Since the valve is switched when the cart has moved and gained momentum equal to its initial momentum, the velocity change is equal to the initial velocity.
Therefore, the acceleration of the cart immediately before the valve is switched is:
acceleration = 0 m/s / time taken = 0 m/s^2
In summary:
(a) The valve should be switched from backward to forward thrust for the cart to end up at rest after the cart has moved a distance and gained momentum equal to its initial momentum.
(b) The mass of the cart remains 430 kg, and its velocity is 0 m/s when the valve is switched.
(c) The magnitude of the thrust is zero since the change in momentum is zero.
(d) The acceleration of the cart immediately before the valve is switched is zero.