Question

If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet per second, then its height h after t seconds is given by the equations h(t) = -16t² 128t (if airresistance is neglected). what is the greatest height the rocket reaches?

Answer 1

The greatest height reached by the toy rocket, launched with an initial velocity of 128 feet per second, is 256 feet. This is found using the vertex of the parabolic equation representing the rocket's height over time.

Step-by-step explanation:

The greatest height a toy rocket can reach when it is launched vertically upwards with an initial velocity can be found using the kinematics equation for projectile motion, specifically the peak of the projectile's flight. The equation given is h(t) = -16t² + 128t. To find the maximum height, we look for the vertex of the parabola represented by this equation, which occurs at the time t when the velocity is zero. This can be found by taking the derivative of the height function and setting it equal to zero, or by using the formula for the vertex of a parabola, t = -b/(2a), where a is the coefficient of t² and b is the coefficient of t. In this case, a = -16 and b = 128, so the rocket reaches its maximum height at t = 128/(2×16) = 4 seconds. Substituting this back into the original equation gives us the maximum height: h(4) = -16(4)² + 128(4) = 256 feet.