Vertical motion with gravity? A rocket is launched vertically upward with an initial velocity of 120 m/s from a platform that is 125 m above the ground. Assume that the only force at work is gravity. Determine and graph the velocity and position functions of the rocket for ?t? ? 0. Then describe the motion in words.
Solution Step 1 Given that the rocket is launched vertically upward with initial velocity 120m/s from a platform of 125 m above the ground. Let us assume that the only force at work is gravity. We have to find the velocity and position function of the rocket for t 0 and to draw the graph of velocity and position function. Step 2 Since the rocket is launched upwards that is against the gravity. Then the initial values for the motion of the object is as follows the acceleration is a = v(t) = 9.8 And v(0) = 120 m/s To find velocity function v(t) , we need to find the antiderivative of v(t) = 9.8 The antiderivative of v(t) = 9.8is v(t) = 9.8t + c Now to find c, applying the initial condition we get, v(0) = 0 + c = c But we have v(0) = 120 Therefore c = 120 Hence the velocity function is v(t) = 9.8t + 120 At maximum height the velocity is zero. That is v(t) = 0 120 9.8t + 120 = 0 t = 9.8 = 12.25 Thus at maximum height t = 12.25 seconds The graph of the velocity function is shown below