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 ay = (2.80 m/s3)t, where the +y-direction is upward. (a) What is the height of the rocket above the surface of the earth at t = 10.0 s?

Verified step by step guidance

1. Identify the given variables: acceleration as a function of time, ay = (2.80 m/s^3) * t, and the time interval, t = 10.0 s.
2. Recognize that the initial velocity (u) is 0 m/s since the rocket starts from rest.
3. Calculate the velocity at any time t by integrating the acceleration function. The velocity function v(t) can be found using the integral of ay with respect to t, plus the constant of integration which is the initial velocity. In this case, integrate ay = (2.80 m/s^3) * t to get v(t) = (1.40 m/s^3) * t^2 + C. Since the initial velocity is 0, C = 0, so v(t) = (1.40 m/s^3) * t^2.
4. Determine the displacement (height) by integrating the velocity function. The height function y(t) can be found by integrating v(t) with respect to t, plus another constant of integration which represents the initial position (y0). Here, integrate v(t) = (1.40 m/s^3) * t^2 to get y(t) = (0.47 m/s^3) * t^3 + C. Since the rocket starts from the surface of the earth, y0 = 0, so C = 0, thus y(t) = (0.47 m/s^3) * t^3.
5. Evaluate y(t) at t = 10.0 s to find the height of the rocket above the surface of the earth at that time.

Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, understanding kinematics is essential to determine the height of the rocket as it moves under a time-dependent acceleration.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In this scenario, the rocket experiences a variable acceleration defined by the equation ay = (2.80 m/s³)t, indicating that the acceleration increases linearly with time. This concept is crucial for calculating the rocket's velocity and position over the given time interval.

Integration

Integration is a mathematical process used to find the total accumulation of a quantity, such as distance or area, from a rate of change. In this context, to find the height of the rocket, we need to integrate the acceleration function to obtain the velocity and then integrate the velocity to find the displacement. This step is vital for solving the problem accurately.