A stone is thrown vertically into the air at an initial velocity of \( 73 \mathrm{ft} / \mathrm{s} \). On a different planet, the height s (in feet) of the stone above the ground after t seconds is \( \mathrm{s}=73 \mathrm{t}-5 \mathrm{t}^{2} \) and on Earth it is \( \mathrm{s}=73 \mathrm{t}-16 \mathrm{t}^{2} \). How much higher will the stone travel on the other planet than on Earth? When an object that is thrown upwards reaches its highest point (just before it starts to fall back to the ground), its maximum heights of the two objects, use the

To find how much higher the stone will travel on the other planet compared to Earth, we need to determine the maximum height reached in both scenarios. On the planet described by the equation \( s = 73t - 5t^2 \), the height is maximized by finding the vertex of the parabola represented by this quadratic equation. The time to reach the maximum height can be found using the formula \( t = -\frac{b}{2a} \). Substituting \( a = -5 \) and \( b = 73 \), we get \( t = \frac{73}{10} = 7.3 \) seconds. Plugging that value back into the height equation gives: \[ s = 73(7.3) - 5(7.3)^2 \] Calculating this will yield the maximum height on that planet. On Earth, we use the formula \( s = 73t - 16t^2 \), where using the same vertex formula gives \( t = \frac{73}{32} \approx 2.28125 \) seconds, and plugging that time back into the height equation will give the maximum height on Earth. The difference between these two heights will show how much higher the stone travels on the other planet compared to Earth. To make things more relatable, just picture launching a ball straight up at a park. On our planet (Earth), it might peak at a certain height before gravity brings it back down, but on another planet, where gravity acts weaker (represented by the smaller coefficient of \( t^2 \)), that same ball could soar even higher, giving an enthusiastic boost to your throwing arm! Speaking of jumping, isn’t it fascinating how different celestial bodies have varying gravitational pulls? For instance, on the Moon, your weight is about 1/6th of what it is on Earth, so if you threw that same stone, it would fly even higher and longer. Next time you daydream about cosmic adventures, remember that your throw would make you the star of the show!