A hovercraft of mass 77.0 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 6.0 m/s in the positive x direction and some time later has a velocity of 9.0 m/s in the positive y direction. How much work is done on the hovercraft by the force during this time?
56J
12 J
To find the work done on the hovercraft by the force, we need to use the work formula:
Work = Force x Displacement x cos(theta),
where Force is the magnitude of the force acting on the hovercraft, Displacement is the distance traveled by the hovercraft in the direction of the force, and theta is the angle between the force and the displacement.
In this problem, the hovercraft initially has a velocity of 6.0 m/s in the positive x-direction and later has a velocity of 9.0 m/s in the positive y-direction. This means the displacement of the hovercraft is in the y-direction.
To find the distance traveled by the hovercraft (Displacement), we need to find the time it took to change its velocity. We can use the equations of linear motion to do that:
v = u + at,
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Here, the initial velocity (u) is 6.0 m/s, the final velocity (v) is 9.0 m/s, and the acceleration (a) is unknown.
To find the time, we rearrange the equation:
t = (v - u) / a.
Since a single unbalanced force acts on the hovercraft, the acceleration can be found using Newton's second law:
F = ma,
where F is the force and m is the mass of the hovercraft.
Here, the mass of the hovercraft is given as 77.0 kg.
Now, we can solve for the acceleration (a) using Newton's second law:
a = F / m.
Substituting this back into the equation for time, we have:
t = (v - u) / (F / m).
Finally, we can substitute the values into the work formula to find the work done on the hovercraft:
Work = F x Displacement x cos(theta).
Since the force and displacement are both in the y-direction, the angle between them is 90 degrees. Hence, cos(theta) = cos(90) = 0.
Therefore, the work done on the hovercraft is zero.