Question

an object of mass 2 kg moves along the x - axis while a net non - linear conservative force is exerted on it. the potential energy of the system consisting of the object as well as other objects is given by the graph. the speed of the object at x = 1 m is 2 m/s. estimate the maximum speed of the object. a 2 m/s b √5 m/s c 2√3 m/s

Explanation:

Step1: Calculate the total mechanical energy

The total mechanical energy \(E\) is the sum of kinetic energy \(K\) and potential energy \(U\).
The kinetic energy at \(x = 1m\) is \(K=\frac{1}{2}mv^{2}\), where \(m = 2kg\) and \(v=2m/s\). So \(K=\frac{1}{2}\times2\times2^{2}=4J\).
From the graph, the potential energy \(U\) at \(x = 1m\) is \(U = 5J\).
Then the total mechanical energy \(E=K + U=4 + 5=9J\).

Step2: Find the minimum potential energy

From the graph, the minimum potential energy \(U_{min}=- 3J\).

Step3: Calculate the maximum kinetic energy

Since \(E = K_{max}+U_{min}\), then \(K_{max}=E - U_{min}\).
Substitute \(E = 9J\) and \(U_{min}=-3J\) into the formula, we get \(K_{max}=9-(-3)=12J\).

Step4: Calculate the maximum speed

We know that \(K_{max}=\frac{1}{2}mv_{max}^{2}\), where \(m = 2kg\) and \(K_{max}=12J\).
So \(\frac{1}{2}\times2\times v_{max}^{2}=12\), \(v_{max}^{2}=12\), \(v_{max}=2\sqrt{3}m/s\).

Answer:

C. \(2\sqrt{3}m/s\)