The position function for a particular object is s = –23t^2 + 65. Which of the following statements is true?
The initial position is 23
The initial velocity is 0
The velocity at time t = 1 is 42
None of these
To determine which of the given statements is true based on the position function, let's break down the equation and the concepts involved.
The position function is given as s = -23t^2 + 65, where "s" represents the position of the object and "t" represents time.
1. The initial position is 23: This statement is not true. The initial position refers to the position of the object at t = 0. We can find the initial position by substituting t = 0 into the position function: s = -23(0)^2 + 65 = 0 + 65 = 65. Therefore, the initial position is 65, not 23.
2. The initial velocity is 0: This statement cannot be determined from the position function alone. The initial velocity refers to the velocity of the object at t = 0. To find the initial velocity, we need the equation for velocity (v) in terms of time (t). The velocity function is the derivative of the position function with respect to time: v = d/dt(-23t^2 + 65). By differentiating the position function, we get v = -46t. Now, substituting t = 0 into the velocity function, we find v = -46(0) = 0. Therefore, the initial velocity of the object is indeed 0.
3. The velocity at time t = 1 is 42: To find the velocity at t = 1, we can again use the velocity function. Substituting t = 1 into the velocity function v = -46t, we get v = -46(1) = -46. Therefore, the velocity at t = 1 is -46, not 42.
Hence, the correct statement is: The initial velocity is 0.