Question

on a distant planet, golf is just as popular as it is on earth. a golfer tees off and drives the ball 4.50 times as far as he would have on earth, given the same initial velocities on both planets. the ball is launched at a speed of 41.1 m/s at an angle of 27.2° above the horizontal. when the ball lands, it is at the same level as the tee. on the distant planet, what are (a) the maximum height and (b) the range of the ball? (a) number units (b) number units attempts: 0 of 3 used submit answer using multiple attempts will impact your score. 20% score reduction after attempt 2 save for later view policies current attempt in progress question 4 of 5

Explanation:

Step1: Analyze vertical - motion for maximum height

The initial vertical velocity is $v_{0y}=v_0\sin\theta$, where $v_0 = 41.1$ m/s and $\theta = 27.2^{\circ}$. So $v_{0y}=41.1\sin(27.2^{\circ})\approx41.1\times0.457 = 18.88$ m/s. The formula for maximum - height in vertical motion is $H=\frac{v_{0y}^{2}}{2g}$. On Earth, $g = 9.8$ m/s². But we first find the relationship between the planets. Since the range on the distant planet is 4.50 times the range on Earth. The range formula is $R=\frac{2v_0^{2}\sin\theta\cos\theta}{g}$. Let $g_E$ be the acceleration due to gravity on Earth and $g_P$ be the acceleration due to gravity on the planet. $\frac{R_P}{R_E}=\frac{g_E}{g_P}=4.50$, so $g_P=\frac{g_E}{4.50}=\frac{9.8}{4.50}\approx2.18$ m/s². Now, using $H=\frac{v_{0y}^{2}}{2g_P}$, we substitute $v_{0y}\approx18.88$ m/s and $g_P\approx2.18$ m/s². $H=\frac{(18.88)^{2}}{2\times2.18}=\frac{356.45}{4.36}\approx81.8$ m.

Step2: Analyze range formula for range

The range formula is $R=\frac{2v_0^{2}\sin\theta\cos\theta}{g}$. We know $v_0 = 41.1$ m/s, $\theta = 27.2^{\circ}$, and $g_P=\frac{9.8}{4.50}$ m/s². $\sin(27.2^{\circ})\approx0.457$, $\cos(27.2^{\circ})\approx0.88$. $R=\frac{2\times(41.1)^{2}\times0.457\times0.88}{9.8/4.50}=\frac{2\times1689.21\times0.40216}{9.8/4.50}=\frac{2\times1689.21\times0.40216\times4.50}{9.8}=\frac{2\times1689.21\times1.80972}{9.8}=\frac{6109.97}{9.8}\approx623.5$ m.

Answer:

(a) 81.8 m
(b) 623.5 m