A rocket starts from rest and moves upward from the surface of the earth. For the first 10.010.010.0 s of its motion, the vertical acceleration of the rocket is given by ay=(2.80a_{y}=(2.80ay=(2.80 m/s3)t)t)t, where the +y+y+y-direction is upward. What is the speed of the rocket when it is 325325 m above the surface of the earth?

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo

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Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problema 49b

Capitolo 2, Problema 49b

A rocket starts from rest and moves upward from the surface of the earth. For the first $10.0$ s of its motion, the vertical acceleration of the rocket is given by $a_{y}=(2.80$ m/s³ $)t$ , where the $+y$ -direction is upward. What is the speed of the rocket when it is $325$ m above the surface of the earth?

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Identify the given information: The rocket starts from rest, so the initial velocity (v0) is 0 m/s. The acceleration is given as a function of time: ay = (2.80 m/s^3)t. We need to find the speed of the rocket when it is 325 m above the surface.

Use the kinematic equation for position with variable acceleration: s = ∫v dt, where v = ∫a dt. First, find the velocity function by integrating the acceleration: v(t) = ∫(2.80 m/s^3)t dt.

Calculate the velocity function: v(t) = (2.80 m/s^3)(t^2)/2 + C. Since the rocket starts from rest, the constant C is 0. Therefore, v(t) = 1.40 m/s^3 * t^2.

Use the position function: s(t) = ∫v(t) dt = ∫(1.40 m/s^3 * t^2) dt. Integrate to find the position function: s(t) = (1.40 m/s^3)(t^3)/3 + C. Since the initial position is 0, C is 0, so s(t) = 0.467 m/s^3 * t^3.

Solve for the time when the rocket is 325 m above the surface: Set s(t) = 325 m and solve for t. Once t is found, substitute it back into the velocity function v(t) = 1.40 m/s^3 * t^2 to find the speed of the rocket at that height.

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Kinematics Equations

Kinematics equations describe the motion of objects without considering the forces that cause the motion. They relate displacement, velocity, acceleration, and time. In this problem, the kinematic equation v = u + at is crucial, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

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Variable Acceleration

Variable acceleration occurs when the acceleration of an object changes with time. In this scenario, the rocket's acceleration is given as a function of time, ay = (2.80 m/s³)t. To find the velocity, we need to integrate this acceleration function with respect to time, considering the initial conditions.

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Integration is a mathematical process used to find quantities like displacement or velocity when given a rate of change, such as acceleration. For this problem, integrating the acceleration function ay = (2.80 m/s³)t with respect to time will yield the velocity function, which can then be evaluated to find the speed at a specific height.

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