At t=0, a particle moving in the x y plane with constant...

At $t=0$, a particle moving in the $x y$ plane with constant acceleration has a velocity of $\mathbf{v}_{i}=(3.00 \mathbf{i}-2.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$ when it is at the origin. At $t=3.00 \mathrm{~s}$, the particle's velocity is $\mathbf{v}=(9.00 \mathbf{i}+7.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$. Find (a) the acceleration of the particle and (b) its coordinates at any time $t$.

At $t=0$, a particle moving in the $x y$ plane with constant acceleration has a velocity of $\mathbf{v}_{i}=(3.00 \mathbf{i}-2.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$
when it is at the origin. At $t=3.00 \mathrm{~s}$, the particle's velocity is $\mathbf{v}=(9.00 \mathbf{i}+7.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$. Find (a) the acceleration of the particle and (b) its coordinates at any time $t$.

Key Concepts

Initial Conditions

Initial conditions, such as the initial position and velocity of the particle, serve as the starting reference point for solving kinematics problems. They are crucial in determining the constants of integration when deriving the equations of motion from acceleration, ensuring that the solutions accurately reflect the motion from the given starting state.

Component-Wise Analysis

Breaking vector quantities into their individual components enables the independent treatment of motion along each axis. This method simplifies problem solving by allowing one to apply one-dimensional kinematics to each direction, and then combine the results to understand the overall motion of the particle.

Vector Representation of Motion

The motion of a particle in the plane is conveniently described using vectors, where each vector encapsulates both magnitude and direction. In problems involving motion, quantities like displacement, velocity, and acceleration are often expressed in vector form to effectively represent their components along perpendicular axes such as x and y.

Kinematic Equations in Two Dimensions

Kinematic equations describe the relationships between displacement, velocity, acceleration, and time for objects moving with constant acceleration. When extended to two dimensions, these equations are applied separately to each axis, allowing the analysis of each component independently while yielding a complete description of the motion.

Transcript

0:00 Hello.

00:01 In this problem, we have a particle that is moving with concessant acceleration in the xy plane.

00:08 At time equals zero, the velocity of the particle was 3 in the i direction and negative 2 in the j direction.

00:17 After 3 seconds, the particle velocity was 9 in the i direction and 7 in the j direction.

00:27 We are first asked to calculate the accelerates or the acceleration vector of this particle, given that it is moving with connoissell acceleration.

00:38 So basically we'll apply the formula that says that the acceleration is just the delta v divided by delta t.

00:48 This is true for both the average acceleration and it is the same as the constant acceleration in case it is constant indeed.

00:58 So let's get this delta v here, which is a very important.

01:02 Vector quantity that is divided by a scalar quantity, which is just the difference in time.

01:09 So delta v here, it's just subtracting this, given that this is the v final and this is the v initial.

01:17 So this will give us to 6 in the direction of i, and then 7 minus minus 2.

01:24 This gives us to 9 in the direction of j.

01:29 Divided by 3, which is the time difference, this gives us 2 in the direction of i and 3 in the direction of j.

01:40 So that's our acceleration vector and that's the answer for part a.

01:46 B, we're asked to determine the coordinates of this particle at any given point in time.

01:55 So applying the equation of kinematics is the same in 2d as, exactly it is in 1d except for that we deal with each axis or each dimension separately when applying those equations so that's just the nature of how vector works so in order to get the position vector which is x in the direction of y and then j y in the direction of j so we need to find both x and y and y so according to the kinematic equations, x is just equal to x dot, which is the initial position, and then the v initial multiplied by t, and finally half a t squared...

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