Question

1. tarzan is swinging on a vine that will break if the force exceeds 2.0 x 103 n. if the length of the vine is 5.0 m and tarzans mass is 1.00 x 102 kg, what is the highest speed he can safely travel while swinging on the vine? (hint: at what point in the swing will the tension in the vine be greatest?)

Explanation:

Step1: Analyze the forces

At the lowest point of the swing, the tension \(T\) in the vine provides the centripetal force \(F_c\) and counteracts Tarzan's weight \(mg\). The centripetal force formula is \(F_c=\frac{mv^{2}}{L}\), and the net force equation is \(T - mg=\frac{mv^{2}}{L}\).

Step2: Rearrange the equation for speed

We want to find the maximum speed \(v\) before the vine breaks. Given \(T = 2.0\times10^{3}\ N\), \(m = 1.00\times10^{2}\ kg\), \(L=5.0\ m\), and \(g = 9.8\ m/s^{2}\). Rearranging \(T - mg=\frac{mv^{2}}{L}\) gives \(v=\sqrt{\frac{(T - mg)L}{m}}\).

Step3: Substitute the values

Substitute \(T = 2.0\times10^{3}\ N\), \(m = 1.00\times10^{2}\ kg\), \(L = 5.0\ m\), and \(g = 9.8\ m/s^{2}\) into \(v=\sqrt{\frac{(T - mg)L}{m}}\).

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Answer:

The highest speed Tarzan can safely travel while swinging on the vine is approximately \(7.1\ m/s\).