Question

An object is launched vertically in the air at 31.85 meters per second from a 9 ​-meter-tall platform. Using the projectile motion model h left parenthesis t right parenthesis equals negative 4.9 t squared plus v 0 t plus h 0 comma where​ h(t) is the height of the projectile t seconds after its​ departure, v 0 is the initial velocity in meters per​ second, and h 0 is the initial height in​ meters, determine how long it will take for the object to reach its maximum height. What is the maximum​ height? a. 2.05 seconds, 15.5 meters b. 1.5 seconds, 20 meters c. 3 seconds, 31.85 meters d. 2.5 seconds , 9 meters

Answer 1

The time it takes for the object to reach its maximum height is 2.051 seconds. The maximum height of the object is approximately 15.48 meters.

Step-by-step explanation:

The time for projectile motion is determined completely by the vertical motion. So, we can use the given equation h(t) = -4.9t^2 + v0t + h0 to determine the time it takes for the object to reach its maximum height. Since the object is launched vertically, the initial horizontal velocity (vx) is 0. We are given that the initial vertical velocity (v0) is 31.85 m/s and the initial height (h0) is 9 m. To find the time it takes for the object to reach its maximum height, we can set the vertical velocity to 0 and solve for t.

Using the equation -4.9t^2 + 31.85t + 9 = 0, we can solve for t using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / 2a. Plugging in the values for a, b, and c, we get t = (-31.85 ± √((31.85)^2 - 4(-4.9)(9))) / (2(-4.9)).

Simplifying the equation, we find two values for t: t = 0.572 seconds and t = 2.051 seconds. Since we are looking for the time it takes to reach the maximum height, we only consider the positive value: t = 2.051 seconds. Therefore, it will take the object 2.051 seconds to reach its maximum height.

To find the maximum height, we can substitute the value of t into the equation h(t) = -4.9t^2 + v0t + h0. Plugging in t = 2.051 seconds, v0 = 31.85 m/s, and h0 = 9 m, we get h(2.051) = -4.9(2.051)^2 + 31.85(2.051) + 9. Simplifying the equation, we find h(2.051) = 15.48 meters. Therefore, the maximum height of the object is approximately 15.48 meters.