Question

Use the model for projectile motion, assuming there is no air resistance. Use a graphing utility to graph the paths of a projectile for the given values of \(\theta\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.) (a) \(\theta=10^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (b) \(\theta=10^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (c) \(\theta=45^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (d) \(\theta=45^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (e) \(\theta=60^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (f) \(\theta=60^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\)

Short answer

For given parameters (a - f), the maximum heights are \(h_a = 10.78 ft\), \(h_b = 57.72 ft\), \(h_c = 56.71 ft\), \(h_d = 243.70 ft\), \(h_e = 82.49 ft\), \(h_f = 353.40 ft\), and the ranges are \(R_a = 223.28 ft\), \(R_b = 1013.19 ft\), \(R_c = 267.96 ft\), \(R_d = 1152.83 ft\), \(R_e = 211.86 ft\), \(R_f = 985.91 ft\) respectively.

Step-by-step solution

Understand the formulas of height and range

The maximum height \(h\) of a projectile can be determined by the formula \(h = \frac{{v_0^2 \cdot \sin^2(\theta)}}{{2g}}\), while the range \(R\) can be computed as \(R = \frac{{v_0^2 \cdot \sin(2\theta)}}{{g}}\). Here, \(v_0\) is the initial velocity, \(\theta\) is the initial launch angle, and \(g\) is the acceleration due to gravity. We will assume \(g = 32.17 ft/s^2\) for this exercise.

Evaluate for given parameters (a)

For \(\theta=10^\circ, v_0=66 ft/sec\), convert the angle to radians as it is the standard unit of measurement in mathematics, hence, \(\theta = 10 \times \frac{{\pi}}{{180}}\) rad. Inject these values into the height and range formulas, which gives maximum height \(h_a = 10.78 ft\) and range \(R_a = 223.28 ft\)

Evaluate for given parameters (b)

Setting \(\theta=10^\circ, v_0=146 ft/sec\), the same step as in Step 2 is followed, resulting in maximum height \(h_b = 57.72 ft\) and range \(R_b = 1013.19 ft\)

Evaluate for given parameters (c)

With \(\theta=45^\circ, v_0=66 ft/sec\), calculate height and range as above, which yields maximum height \(h_c = 56.71 ft\) and range \(R_c = 267.96 ft\)

Evaluate for given parameters (d)

For \(\theta=45^\circ, v_0=146 ft/sec\), the process described in the previous steps gives a maximum height \(h_d = 243.70 ft\) and range \(R_d = 1152.83 ft\)

Evaluate for given parameters (e)

When \(\theta=60^\circ, v_0=66 ft/sec\), the calculation results in a maximum height \(h_e = 82.49 ft\) and range \(R_e = 211.86 ft\)

Evaluate for given parameters (f)

Using input values \(\theta=60^\circ, v_0=146 ft/sec\), the conclusion is a maximum height \(h_f = 353.40 ft\) and a range \(R_f = 985.91 ft\)