The Correct Answer and Explanation is
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- In considering the general case of non-uniform acceleration, it is not always possible to express position as a simple function of time. However, consider the specific case of an object whose acceleration is given by the equation a(t) = (1/3)t, where a is in meters per second squared and t is in seconds. The object starts from rest at the origin at t = 0. Find the object’s velocity and position as functions of time.
- The position of a particle moving along the x-axis is given by x(t) = 3.0t^2 – 2.0t^3, where x is in meters and t is in seconds.
(a) What is the position of the particle at t = 2.0 s?
(b) What is the instantaneous velocity of the particle at t = 2.0 s?
(c) What is the instantaneous acceleration of the particle at t = 2.0 s?
(d) At what time(s) is the particle at rest? - A particle moves along the x-axis. Its position is given by the equation x = 2 + 3t – 4t^2, with x in meters and t in seconds. Determine (a) its position at the instant it changes direction, and (b) its velocity when it returns to the position it had at t = 0.
- Two cars, A and B, move along the x-axis. Car A starts from rest at t = 0 with a constant acceleration of 2.5 m/s^2. Car B passes the origin at t = 0 with a constant velocity of 20 m/s.
(a) Find the time at which they have the same position.
(b) Find the position at that time.
(c) Find their velocities at that time.
The provided image contains four physics problems focused on the topic of one dimensional kinematics, which is the study of motion. These questions are typical of a high school or introductory university physics course, designed to test a student’s understanding of the relationships between position, velocity, and acceleration.
Problem 10 introduces a scenario with non uniform acceleration that varies linearly with time. To solve this, one must use calculus, specifically integration, to derive the velocity function from the given acceleration function, and then integrate the velocity function to find the position function, applying the initial conditions that the object starts from rest at the origin.
Problem 11 provides a cubic function for a particle’s position. It requires finding the particle’s position, velocity, and acceleration at a specific moment. This involves evaluating the position function at a given time and calculating the first and second derivatives of the position function to find the velocity and acceleration functions, respectively. The final part asks when the particle is at rest, which means finding the time when its velocity is zero.
Problem 12 presents another quadratic position function. It asks for the particle’s position when its velocity is zero (when it changes direction) and its velocity when it returns to its initial position. This combines differentiation to find the velocity with algebraic problem solving.
Finally, problem 13 is a classic relative motion problem involving two objects with different motion profiles: one with constant acceleration and another with constant velocity. Solving it involves setting up the position equations for both cars and finding the time at which their positions are equal. This leads to solving a quadratic equation to find when and where they meet.
