You have a remote-controlled car that has been programmed to have velocity $v=(-3ti+2t^{2}j)\text{m/s}$, where t is in s. At t = 0 s, the car is at $r_0=(3.0i+2.0j)\text{m}$. What are the car's position vector?

Verified step by step guidance

1. Step 1: Understand the problem. The velocity vector of the car is given as v = (3t i + 2t^2 j) m/s, and the initial position vector at t = 0 s is r0 = (3.0 i + 2.0 j) m. To find the position vector r(t) at any time t, we need to integrate the velocity vector with respect to time and add the initial position vector.
2. Step 2: Write the expression for the position vector r(t). The position vector is obtained by integrating the velocity vector v(t) with respect to time: r(t) = ∫v(t) dt + r0. Substitute v(t) = (3t i + 2t^2 j) into the integral.
3. Step 3: Perform the integration for each component of the velocity vector. For the i-component: ∫3t dt = (3/2)t^2. For the j-component: ∫2t^2 dt = (2/3)t^3. Combine these results to form the position vector: r(t) = [(3/2)t^2 i + (2/3)t^3 j] + r0.
4. Step 4: Add the initial position vector r0 to the integrated result. The initial position vector is r0 = (3.0 i + 2.0 j). Add this to the integrated position vector: r(t) = [(3/2)t^2 i + (2/3)t^3 j] + (3.0 i + 2.0 j).
5. Step 5: Combine the terms to express the final position vector. Group the i and j components: r(t) = [(3.0 + (3/2)t^2) i + (2.0 + (2/3)t^3) j]. This is the position vector of the car at any time t.

Key Concepts

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 case, the velocity of the remote-controlled car is given as a function of time, showing how its speed and direction change over time.

Position Vector

The position vector represents the location of an object in space relative to a defined origin. It is expressed in terms of its components along the coordinate axes, typically denoted as r = xi + yj + zk. For the remote-controlled car, the position vector at any time t can be determined by integrating the velocity function over time, starting from the initial position.

Integration in Physics

Integration is a mathematical process used to find the total accumulation of a quantity, such as distance traveled over time. In physics, when calculating position from velocity, integration allows us to determine how far an object has moved from its initial position. For the car's motion, integrating the velocity function with respect to time will yield the position vector as a function of time.