A hovercraft of mass 69.0 kg can move on a horizontal surface, the x-y plane. A single unbalanced force acts on the hovercraft, but the size of the force is unknown. The hovercraft initially has a velocity of 5.0 m/s in the positive x direction and some time later has a velocity of 2.0 m/s in the positive y direction. How much work is done on the hovercraft by the force during this time?
I'm having trouble visualizing and applying the appropriate work formula.
The work done is what increases its kinetic energy, and equals the amount of that increase. (In this case, it is a decrease, so negative work is done). The directions are not important. Just compute the change in kinetic energy.
W = delta(KE) = -(1/2)(69)(25 - 4)
= -724.5 J
To calculate the work done on an object, you need to know the force applied to it and the displacement it undergoes. In this case, we know the initial and final velocities of the hovercraft but not the force acting on it.
To find the work done, we need to calculate the change in kinetic energy of the hovercraft. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:
Work = Change in Kinetic Energy
The kinetic energy of an object can be calculated using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
Initially, the hovercraft has a velocity of 5.0 m/s in the positive x direction. Its initial kinetic energy can be calculated as:
Initial Kinetic Energy = (1/2) * mass * (initial velocity)^2
Once we know the initial and final velocities, we can calculate the change in kinetic energy using the formula:
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Now, let's calculate the initial kinetic energy:
Initial Kinetic Energy = (1/2) * 69.0 kg * (5.0 m/s)^2
Simplifying, we get:
Initial Kinetic Energy = 862.5 J
Given that the final velocity is 2.0 m/s in the positive y direction, we can calculate the final kinetic energy:
Final Kinetic Energy = (1/2) * 69.0 kg * (2.0 m/s)^2
Simplifying, we get:
Final Kinetic Energy = 138.6 J
Now, let's calculate the change in kinetic energy:
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Change in Kinetic Energy = 138.6 J - 862.5 J
Simplifying, we get:
Change in Kinetic Energy = -723.9 J
Since the change in kinetic energy is negative, it means that work is done on the hovercraft.
Therefore, the work done on the hovercraft by the force during this time is 723.9 Joules.
To determine the work done on the hovercraft, we need to recall the formula for work:
Work (W) = Force (F) * Distance (d) * cos(theta)
In this case, the force acting on the hovercraft is unbalanced and unknown, and the distance traveled is also unknown. However, we can still find the work done by the force using the concept of work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In other words:
Work (W) = Change in Kinetic Energy
Now, we can find the change in kinetic energy of the hovercraft.
The initial kinetic energy of the hovercraft can be calculated using the formula:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
KE_initial = (1/2) * 69.0 kg * (5.0 m/s)^2
Similarly, the final kinetic energy can be calculated using the formula:
KE_final = (1/2) * 69.0 kg * (2.0 m/s)^2
Now, the change in kinetic energy (ΔKE) is simply the difference between the final and initial kinetic energies.
ΔKE = KE_final - KE_initial
Finally, the work done on the hovercraft is equal to the change in kinetic energy:
Work (W) = ΔKE
You can substitute the values into the equations and calculate the final answer.
Remember, work is a scalar quantity, so it can be either positive or negative, depending on the direction of the force and the resulting change in kinetic energy.