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?
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.
Please explain to me how to integrate-I am unsure exactly what this means in this problem
Thanks
Find the area under the line within each interval
To find the velocity of the particle at 2 cm and 4 cm, you can use Newton's second law of motion: F = m*a, where F is the force, m is the mass, and a is the acceleration.
First, let's determine the acceleration at each point on the graph. From the graph, you can see that the force acting on the particle changes linearly with the velocity. At 0 cm, the force is 10 N, and at 4 cm, the force is -10 N. The change in force over the 4 cm displacement is 10 N - (-10 N) = 20 N.
The mass of the particle is given as 1.6 kg. To find the acceleration, we can use the formula a = F/m. Thus, a = (20 N) / (1.6 kg) = 12.5 m/s^2.
Since the acceleration is constant, we can use the formula for uniformly accelerated motion to find the velocity. The formula is v = u + a*t, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
At x = 0 cm, the particle's velocity is given as 4.2 m/s. Plugging this value into the formula, we obtain the equation:
v = 4.2 m/s + (12.5 m/s^2)(t)
To find the velocity at x = 2 cm, we need to find the time it takes for the particle to travel that distance. The displacement is given by x = u*t + 0.5*a*t^2, where x is the displacement, u is the initial velocity, a is the acceleration, and t is the time. In this case, x = 2 cm = 0.02 m. Rearranging the equation, we have:
0.02 m = 4.2 m/s * t + 0.5 * 12.5 m/s^2 * t^2
Simplifying the equation, we get a quadratic equation in terms of t:
0.02 m = 4.2 m/s * t + 6.25 m/s * t^2
Solving this quadratic equation will give us the value of t, which we can then substitute into the velocity equation to find the velocity at x = 2 cm.
Similarly, to find the velocity at x = 4 cm, we can use the same approach. The displacement is now x = 4 cm = 0.04 m. Solving the resulting quadratic equation will give us the value of t, which can be substituted into the velocity equation to find the velocity at x = 4 cm.