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Chapter 2: Problem 310

\(\mathrm{t}_{1}\) and \(\mathrm{t}_{2}\) are two values of time of a projectile at the same height \(t \mathrm{t}_{1}+\mathrm{t}_{2}=\) (A) Time to reach maximum height (B) flight time for the projectile (C) \((3 / 4)\) time of the flight time. (D) \((3 / 2)\) time of the flight time.

Short Answer

Expert verified
The relationship between \(t_1\) and \(t_2\) is given by \(t_1 + t_2 = \frac{2v_0\sin\theta}{g}\), where \(v_0\) is the initial velocity, \(\theta\) is the launch angle, and \(g\) is the acceleration due to gravity. This expression represents the total flight time of a projectile, making the correct answer (B) flight time for the projectile.

Step by step solution

01

Write down the equation for the height of a projectile as a function of time

The height of a projectile at a given time can be represented as: \[ h(t) = h_{0} + v_{0}\sin\theta \cdot t - \frac{1}{2}gt^2 \] where - \(h_{0}\) is the initial height (we can assume it to be 0 for simplicity), - \(v_{0}\) is the initial velocity, - \(\theta\) is the launch angle, - \(g\) is the acceleration due to gravity (\(9.81\,m/s^{2}\)), and - \(t\) is the time. With \(h_{0} = 0\), the equation becomes: \[ h(t) = v_{0}\sin\theta \cdot t - \frac{1}{2}gt^2 \] Since \(h(t_1) = h(t_2)\) we have: \[ v_{0}\sin\theta \cdot t_1 - \frac{1}{2}gt_1^2 = v_{0}\sin\theta \cdot t_2 - \frac{1}{2}gt_2^2 \]
02

Solve for \(t_1 + t_2\) from the equation

By solving the equation we got earlier, we can get the value of \(t_1 + t_2\). Let's rearrange the equation to get \(t_1 + t_2\) as follows: \[ v_{0}\sin\theta \cdot (t_1 - t_2) = \frac{1}{2}g(t_2^2 - t_1^2) \] \[ v_{0}\sin\theta \cdot (t_1 - t_2) = \frac{1}{2}g(t_2 + t_1)(t_2 - t_1) \] Now, dividing both sides by \((t_1 - t_2)\), we get: \[ v_{0}\sin\theta = \frac{1}{2}g(t_2 + t_1) \] Now, solving for \(t_1 +t_2\): \[ t_1 + t_2 = \frac{2v_0\sin\theta}{g} \]
03

Identify the relationship between \(t_1\) and \(t_2\)

From Step 2, we derived that \(t_1 + t_2 = \frac{2v_0\sin\theta}{g}\). This is the expression for the total flight time of a projectile launched at an angle \(\theta\) with an initial velocity \(v_0\). Therefore, the correct option is: (B) flight time for the projectile

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