10. [-/1 Points] DETAILS MY NOTES SCALCET9 2.XP.7.008.MI. \( \begin{array}{l}\text { A ball is thrown into the air with a velocity of } 37 \mathrm{f} / \mathrm{s} \text {. Its height, in feet, after } t \text { seconds is given by } s(t)=37 t-16 t^{2} \text {. Find } \\ \text { the velocity (in } \mathrm{ft} / \mathrm{s} \text { ) of the ball at time } t=1 \text { second. } \\ \mathrm{ft} / \mathrm{s}\end{array} \)

To find the velocity of the ball at \( t = 1 \) second, we first need to find the derivative of the height function \( s(t) = 37t - 16t^2 \). The derivative \( s'(t) \) represents the velocity of the ball. Calculating the derivative, we get: \[ s'(t) = 37 - 32t \] Now, substitute \( t = 1 \): \[ s'(1) = 37 - 32(1) = 37 - 32 = 5 \text{ ft/s} \] So, the velocity of the ball at \( t = 1 \) second is \( 5 \text{ ft/s} \). The concept of projectile motion has been around for centuries, captivating thinkers from Aristotle to Galileo. They laid the groundwork by studying how objects behave when thrown into the air. By using simple equations, we can now predict how high and how fast our thrown ball travels, maximizing our understanding of motion! In real life, understanding this projectile motion isn't just for physics geeks; it applies to sports! Basketball players aim for the hoop, while soccer players kick the ball with precision. By knowing how velocity and height interplay, athletes can enhance their skills and nail that perfect shot every time.