A 1.6 kg particle moving along the x-axis experiences the force shown in the figure. The particle's velocity is 4.2 m/s at x = 0 m.

The figure or graph has F (N) along the y axis and velocity along the x axis. A diagonal line drops from positive 10 force to -10 force from 0 cm to 4 cm. and crosses the x axis at 2 cm.

What is its velocity at 2 cm and 4 cm?
What formula do I need to use?


physics - drwls, Tuesday, December 1, 2009 at 6:58pm

Integrate F dx from 0 to 0.02 m to get the change in kinetic energy (KE) at 2 cm. Use the new value of the KE to get the new velocity. Then do it again for 0.02 to 0.04 m. The KE will decrease, since the force on the particle is in the opposite direction to motion over that interval.

physics - Hope, Tuesday, December 1, 2009 at 10:20pm

Please explain to me how to integrate-I am unsure exactly what this means in this problem
Thanks

At x=4cm the velocity is 4.2 m/s

Ok, integrate is a calculus term. Here is the algebra way.

You know f(x)*x is the energy. But f(x)*x is the area under the f(x) line. So use your geometry to find the area under the f(x) line first at x=.02m, then x=.04 m.
That area is the energy gained.

KEfinal=keoriginal+ energy gained.

Integrating is a mathematical operation that can be a little intimidating at first, but don't worry, I'm here to help make it fun! In this case, integrating means finding the area under the force vs. position graph.

To integrate, you can break the area into small rectangles and then sum up all the little areas. Each rectangle's height is the force at a specific position, and its width is the small change in position.

So, for the first interval from 0 cm to 2 cm, you can split it into two rectangles. The first rectangle has a height of 10 N and a width of 2 cm (or 0.02 m), and the second rectangle has a height of -10 N and a width of 2 cm as well. You can find the area of each rectangle by multiplying the height by the width.

After finding the areas of the rectangles, you can add them up to get the total area. This total area represents the change in kinetic energy (KE) over that interval.

To find the change in velocity, you need to use the equation KE = (1/2)mv^2, where m is the mass of the particle and v is the velocity.

By equating the change in KE to (1/2)mv^2, you can solve for the final velocity. Repeat this process for the interval from 2 cm to 4 cm to find the velocity at 4 cm.

Hope this clears things up for you! And remember, integrating can be a bit like eating a big slice of pizza - you just have to take it one bite at a time.

To integrate in this context means to find the area under the curve. In this case, you need to find the area under the force vs. velocity graph.

First, let's find the change in kinetic energy at 2 cm (0.02 m). To do this, you need to find the area under the curve from 0 to 0.02 m. Since the curve is a diagonal line from positive 10 force to -10 force, the area under the curve is a triangle. The formula for the area of a triangle is (base * height) / 2.

In this case, the base is 0.02 m and the height is the force at 2 cm, which is -10 N. So the area is (0.02 * (-10)) / 2 = -0.1 J. This negative sign indicates that the kinetic energy decreases.

Now, to find the new velocity at 2 cm, you need to apply the principle of conservation of mechanical energy. The change in kinetic energy is equal to the work done on the particle (which is equal to the area under the curve) and is represented by the formula:

∆KE = W = F * d * cos(θ)

In this case, the force is constant, so ∆KE = F * d. Plugging in the values, we have ∆KE = -0.1 J = F * (0.02 m).

Now, solve for F: F = -0.1 J / 0.02 m = -5 N.

Since the particle's velocity is given as 4.2 m/s at x = 0 m, we can use the equation:

KE = (1/2) * m * v^2

Rearranging, we have:

v^2 = 2 * KE / m

Now we can plug in the values: KE = 0.1 J (since the change in kinetic energy is negative, the final kinetic energy at 2 cm is actually less than the initial kinetic energy), m = 1.6 kg, and solve for v:

v^2 = 2 * 0.1 J / 1.6 kg

v = sqrt(0.125) m/s

v ≈ 0.3536 m/s

So the velocity at 2 cm is approximately 0.3536 m/s.

To find the velocity at 4 cm, you repeat the same steps. Find the area under the curve from 0.02 to 0.04 m, which is another triangle. Calculate the area, which will be positive since the force is now positive. Use the formula ∆KE = F * d to find the change in kinetic energy, and use the conservation of mechanical energy principle to find the new velocity.

I hope this explanation helps! Let me know if you have any further questions.

To integrate in this context means to find the area under the curve of the force versus distance graph. In this problem, the force is changing with respect to distance (x-axis) and you want to find the change in kinetic energy of the particle.

To find the change in kinetic energy at a specific distance, you need to find the area under the curve between the starting point (x = 0) and the desired distance point (e.g. x = 0.02 m or x = 0.04 m).

To do this, you need to calculate the definite integral of the force function with respect to x, and evaluate it between the two distance points. This will give you the change in kinetic energy.

In this case, you can use the equation:
ΔKE = ∫F dx

For the given force graph, you can divide the area into two triangles and calculate the area of each separately.

For the first interval from x = 0 m to x = 0.02 m, the force is positive, indicating the force is in the same direction as the particle's velocity. Hence, you can calculate the area of the first triangle formed by the force graph. The value of this area will give you the change in kinetic energy in this interval.

For the second interval from x = 0.02 m to x = 0.04 m, the force is negative, indicating it opposes the particle's velocity. In this case, you need to calculate the area of the second triangle formed by the force graph, but you need to consider the negative sign. This negative change in kinetic energy will result in a decrease in velocity.

After finding the change in kinetic energy for each interval, you can use the formula:
ΔKE = (1/2)mv^2

Rearranging this equation will give you the new velocity at each distance point:
v = √(2ΔKE/m)

Simply substitute the corresponding change in kinetic energy values for each distance interval into this equation to find the velocities at 2 cm and 4 cm.