A 100-N force has a horizontal component of 80 N and a vertical component of 60 N. The force is applied to a box which rests on a level frictionless floor. The cart starts from rest, and moves 2.0 m horizontally along the floor. What is the cart's final kinetic energy?
workin= horizontalforce*distance=80*2=160J
final ke: 160 J
Well, the horizontal component of the force does work on the box, since it's the force in the same direction as the displacement. The work done, W, can be calculated using W = F * d * cos(theta), where F is the force, d is the displacement, and theta is the angle between the force and the displacement. In this case, theta is 0 degrees since the force and displacement are in the same direction.
So, W = 80 N * 2.0 m * cos(0 degrees) = 160 J
Now, the work done is equal to the change in kinetic energy, so the final kinetic energy, K, is 160 J.
But let's not forget about the vertical component of the force! It doesn't do any work since it's perpendicular to the displacement. So despite its best efforts to contribute, the vertical component just ends up adding a splash of humor to the situation.
To find the cart's final kinetic energy, we need to calculate the work done on the cart.
The work done on an object is given by the equation:
work = force * distance * cos(theta)
In this case, the force is the horizontal component of the applied force, which is 80 N.
The distance is the horizontal distance the cart moves, which is 2.0 m.
And theta is the angle between the applied force and the direction of motion, which is 0 degrees (since the applied force is horizontal).
So, the work done on the cart is:
work = 80 N * 2.0 m * cos(0)
cos(0) is equal to 1, so the work done on the cart is:
work = 80 N * 2.0 m * 1
work = 160 J
The work done on the cart is equal to the change in kinetic energy of the cart, so the cart's final kinetic energy is 160 J.
To find the cart's final kinetic energy, we need to calculate its final speed first. We can use the horizontal component of the force to do this.
The horizontal component of the force is responsible for the cart's horizontal acceleration. We can use Newton's second law of motion, which states that force equals mass times acceleration (F = ma). In this case, the horizontal force is the horizontal component of 80 N, and the mass of the cart is unknown.
Since the cart rests on a level floor and there is no friction, the vertical component of the force does not affect the cart's motion horizontally. Therefore, we can ignore it in our calculations.
Using Newton's second law, we can rearrange the equation to solve for acceleration (a).
a = F/m
Now, we know the horizontal component of the force is 80 N. However, we do not know the mass of the cart.
To find the cart's mass, we need to use the gravitational force formula:
F_gravity = mg
The vertical component of the force (60 N) is equal to the gravitational force on the cart.
So, 60 N = mg
We can rearrange the equation to solve for m (mass):
m = 60 N / g
Substituting this value of m back into the horizontal force equation, we can calculate the cart's horizontal acceleration (a).
Now that we have the cart's horizontal acceleration, we can use one of the equations of motion to find the final speed. Since the cart started from rest and we know the displacement (2.0 m), we can use the equation:
v^2 = u^2 + 2as
where u is the initial velocity (which is 0 m/s), and s is the displacement (2.0 m).
We can rearrange the equation to solve for the final velocity (v):
v = sqrt(2as)
Now that we have the final velocity, we can calculate the cart's final kinetic energy.
The kinetic energy of an object is given by the formula:
KE = 0.5mv^2
where m is the mass and v is the velocity. Substituting the known values, we can calculate the cart's final kinetic energy.