Question

The force Fx acting on a 0.500-kg particle is shown as a function of x in (see below). (a) From the graph, calculate the work done by the force when the particle moves from x = 0.00 to the following values of x: -4.00, -3.00,-2.00, -1.00, +1.00, +2.00, +3.00, and +4.00 m. (b) If it starts with a velocity of 2.00 m/s in the +x direction, how far will the particle go in that direction before stopping?

Answer 1

Equation: Kinetic Energy final- Kinetic energy initial = delta kinetic energy.


Kinetic Energy Initial= 1/2mv^2= 1/2(0.5)(2.0)^2= 1.00 Joules of starting energy.

Delta Kinetic Energy = 1/2(mass)(final velocity)^2- 1/2m(initial velocity)^2

Either way, we want the Kinetic Energy to reach zero.

This particle needs to lose that kinetic energy before it can stop...so it must lose 1 Joule of kinetic energy.

To find the remaining Joules, find the area under the curve (because our graph is in relation of Force and distance and force*distance= Work in Joules. (1 Joule equals one square). Half a Joule equals half a square.

The Joules below the x axis are negative because of the above reason and the Joules above the x-axis are positive because of the above reason.

The slope of the graph switches between rising 2N/meters and -2N/Meters and flatlining

The area of the first gap from 0 to 1 meter is a triangle like:

Area= ((b)(h)) /2 = (2*1)/2 =1.0 Joules.

So kinetic energy starting 1.0 Joules =2.00 Joules now, but we need to lose 2.00 Joules now so that the particle can stop at zero kinetic energy.

Then from 1 meter to 2 meters is another area equation:

Area= (b*h)/2 = -(2*1)/2 = -1.0 Joules....

Now Kinetic energy at this point is 2.0 Joules-1.00 Joules= 1.0 Joules. Now we just need to lose one more joule before the particle will stop.

And if we go from x=2 meters to x=3 meters then

Area of rectangle= -(2*1) =-2 Joules...so now we have...

1.0 Joules -2.00 Joules=-1.00 Joules.

Well, the particle must have stopped somewhere between x=2 meters and x=3 meters. If we calculate the area halfway in-between we find that the Kinetic energy in Joules reaches stopping distance at 2.5 meters having 0 Joules of Kinetic energy for motion.

=2.5 meters