You have a remot-controlled car that has been programmed to have velocity v = (3ti + 2t^2j) m/s, where t is in s. At t = 0 s, the car is at r0 = (3.0i + 2.0j) m. What are the car's (a) position vector
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Step 1: Understand the given velocity vector, v = (3ti + 2t^2j) m/s. This expression shows how the velocity components in the i (horizontal) and j (vertical) directions change with time t.
Step 2: To find the position vector at any time t, integrate the velocity vector with respect to time. The position vector r(t) can be found by integrating each component of the velocity vector separately. The integration will be r(t) = \int v(t) dt + r_0, where r_0 is the initial position vector.
Step 3: Integrate the i-component of the velocity vector, 3t, with respect to t. This will give you the i-component of the position vector. Remember to add the i-component of the initial position vector after integration.
Step 4: Integrate the j-component of the velocity vector, 2t^2, with respect to t. This will give you the j-component of the position vector. Remember to add the j-component of the initial position vector after integration.
Step 5: Combine the integrated i and j components to form the complete position vector r(t) = (integrated i-component)i + (integrated j-component)j. This vector represents the position of the car at any time t.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity
Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction, which means it indicates how fast an object is moving and in which direction. In this question, the velocity of the remote-controlled car is given as a function of time, which will be essential for determining its position over time.
The position vector represents the location of an object in space relative to a reference point, typically the origin of a coordinate system. It is expressed in terms of its components along the coordinate axes. In this case, the position vector of the car can be calculated by integrating the velocity function over time, starting from the initial position provided.
Integration is a fundamental concept in calculus that allows us to find the accumulated value of a quantity over an interval. In physics, it is often used to determine position from velocity by integrating the velocity function with respect to time. This process will be necessary to find the car's position vector at any given time based on its velocity function.