Starting from origin at t=0, with initial velocity `5 hat(j) m//s`, a particle moves in the `x-y` plane with a constant acceleration of `(10 hat(i) - 5hat(j))m//s^(2)`. Find the coordinates of the particle at the moment its y-coordinate is maximum.

Options

A. (5m, 5m)
B. (2.5m, 2.5m)
C. (5m, 2.5m)
D. (2.5m, 5m)

Explanation

To solve the problem step by step, we will use the equations of motion to find the coordinates of the particle when its y-coordinate is maximum.

### Step 1: Identify the initial conditions and acceleration
- The initial position of the particle is at the origin:
\[ \vec{r_0} = 0 \hat{i} + 0 \hat{j} \text{ m} \]
- The initial velocity is:
\[ \vec{u} = 0 \hat{i} + 5 \hat{j} \text{ m/s} \]
- The acceleration is:
\[ \vec{a} = 10 \hat{i} - 5 \hat{j} \text{ m/s}^2 \]

### Step 2: Write the equations of motion
Using the second equation of motion, the displacement vector \(\vec{s}\) can be expressed as:
\[ \vec{s} = \vec{u} t + \frac{1}{2} \vec{a} t^2 \]
Substituting the values of \(\vec{u}\) and \(\vec{a}\):
\[ \vec{s} = (0 \hat{i} + 5 \hat{j}) t + \frac{1}{2} (10 \hat{i} - 5 \hat{j}) t^2 \]
This simplifies to:
\[ \vec{s} = (0 + 5t) \hat{j} + \left(5t^2\right) \hat{i} - \left(\frac{5}{2} t^2\right) \hat{j} \]
Combining the terms, we have:
\[ \vec{s} = 5t^2 \hat{i} + \left(5t - \frac{5}{2} t^2\right) \hat{j} \]

### Step 3: Identify the coordinates
From the displacement vector, we can identify the x and y coordinates:
- \(x_f = 5t^2\)
- \(y_f = 5t - \frac{5}{2} t^2\)

### Step 4: Find the time when y-coordinate is maximum
To find the maximum y-coordinate, we differentiate \(y_f\) with respect to time \(t\):
\[ \frac{dy_f}{dt} = 5 - 5t \]
Setting the derivative equal to zero to find the critical points:
\[ 5 - 5t = 0 \implies t = 1 \text{ s} \]

### Step 5: Verify that it is a maximum
To confirm that this point is a maximum, we differentiate \(\frac{dy_f}{dt}\) again:
\[ \frac{d^2y_f}{dt^2} = -5 \]
Since \(\frac{d^2y_f}{dt^2} < 0\), this indicates that \(y_f\) is indeed at a maximum when \(t = 1\) s.

### Step 6: Calculate the coordinates at \(t = 1\) s
Substituting \(t = 1\) s into the equations for \(x_f\) and \(y_f\):
- For \(x_f\):
\[ x_f = 5(1)^2 = 5 \text{ m} \]
- For \(y_f\):
\[ y_f = 5(1) - \frac{5}{2}(1)^2 = 5 - 2.5 = 2.5 \text{ m} \]

### Final Coordinates
The coordinates of the particle when its y-coordinate is maximum are:
\[ (x_f, y_f) = (5 \text{ m}, 2.5 \text{ m}) \]