A force in the +x-direction with magnitude F(x)=18.0N−(0.530N/m)x is applied to a 7.60kg box that is sitting on the horizontal, frictionless surface of a frozen lake. F(x) is the only horizontal force on the box.
If the box is initially at rest at x=0, what is its speed after it has traveled 17.0m ?
v=?
answer this please
That was wrong
7.77m/s
To find the speed of the box after it has traveled a distance of 17.0m, we can use the work-energy theorem. The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
The work done on an object is given by the integral of the force applied over the distance traveled:
W = ∫ F(x) dx
In this case, the force applied is given by F(x) = 18.0N - (0.530N/m)x. We can substitute this expression into the work equation:
W = ∫ (18.0N - (0.530N/m)x) dx
To find the speed of the box, we need to calculate the net work done on the box. In this case, the net work is equal to the work done by the external force F(x).
The work done by a force is defined as the force multiplied by the displacement in the direction of the force:
W = F(x) * Δx
where Δx is the change in position. Since the box is initially at rest, its initial position is x = 0. Therefore, we can write:
W = F(x) * (xf - xi)
where xi is the initial position and xf is the final position.
In this case, xi = 0m and xf = 17.0m. So the work done by the force F(x) is:
W = F(x) * (17.0m - 0m)
Now we can substitute the expression for F(x) and solve for W:
W = (18.0N - (0.530N/m)(17.0m)) * 17.0m
W = (18.0N - 9.01N) * 17.0m
W = 8.99N * 17.0m
W = 152.83 J (joules)
Since the work done is equal to the change in kinetic energy, we set W equal to the change in kinetic energy:
W = ΔKE
ΔKE = 152.83 J
The change in kinetic energy is given by:
ΔKE = KEf - KEi
where KEi is the initial kinetic energy and KEf is the final kinetic energy. Since the box is initially at rest, its initial kinetic energy is zero.
ΔKE = KEf - 0
ΔKE = KEf
Therefore, the final kinetic energy of the box is equal to 152.83 J. We can use the kinetic energy formula to find the speed of the box:
KEf = (1/2) * m * v^2
where m is the mass of the box in kilograms and v is the speed of the box in meters per second.
Rearranging the equation and solving for v, we get:
v = sqrt((2 * KEf) / m)
Substituting the values, we get:
v = sqrt((2 * 152.83 J) / 7.60 kg)
v = sqrt(305.66 J / 7.60 kg)
v = sqrt(40.1973684 m^2/s^2)
v = 6.34 m/s
Therefore, the speed of the box after it has traveled 17.0m is 6.34 m/s.