A particle is moving in x-y p1ane.At time t`=0,` particle is at `(1m, 2m)` and has velocity `(4 hat i + 6 hat j) m//s.` At `t = 4s,` particle reaches at `(6m, 4m)` and has velocity `(2 hat i + 10 hat j) m//s.` In the given time interval, find
(a) average velocity,
(b) average acceleration and
(c) from the given data, can you find average speed?

Explanation

To solve the problem step by step, we will follow the instructions given in the question and calculate the average velocity, average acceleration, and check if we can find the average speed.

### Given Data:
- Initial position at \( t = 0 \): \( \mathbf{r}_0 = (1 \, \hat{i} + 2 \, \hat{j}) \, \text{m} \)
- Initial velocity at \( t = 0 \): \( \mathbf{v}_0 = (4 \, \hat{i} + 6 \, \hat{j}) \, \text{m/s} \)
- Final position at \( t = 4 \): \( \mathbf{r}_f = (6 \, \hat{i} + 4 \, \hat{j}) \, \text{m} \)
- Final velocity at \( t = 4 \): \( \mathbf{v}_f = (2 \, \hat{i} + 10 \, \hat{j}) \, \text{m/s} \)

### (a) Average Velocity
The average velocity \( \mathbf{v}_{avg} \) is given by the formula:
\[ \mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t} \]
where \( \Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_0 \) and \( \Delta t = t_f - t_0 \).

1. Calculate \( \Delta \mathbf{r} \):
\[ \Delta \mathbf{r} = (6 \, \hat{i} + 4 \, \hat{j}) - (1 \, \hat{i} + 2 \, \hat{j}) = (6 - 1) \, \hat{i} + (4 - 2) \, \hat{j} = 5 \, \hat{i} + 2 \, \hat{j} \]

2. Calculate \( \Delta t \):
\[ \Delta t = 4 \, \text{s} - 0 \, \text{s} = 4 \, \text{s} \]

3. Now calculate \( \mathbf{v}_{avg} \):
\[ \mathbf{v}_{avg} = \frac{5 \, \hat{i} + 2 \, \hat{j}}{4} = 1.25 \, \hat{i} + 0.5 \, \hat{j} \, \text{m/s} \]

### (b) Average Acceleration
The average acceleration \( \mathbf{a}_{avg} \) is given by the formula:
\[ \mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t} \]
where \( \Delta \mathbf{v} = \mathbf{v}_f - \mathbf{v}_0 \).

1. Calculate \( \Delta \mathbf{v} \):
\[ \Delta \mathbf{v} = (2 \, \hat{i} + 10 \, \hat{j}) - (4 \, \hat{i} + 6 \, \hat{j}) = (2 - 4) \, \hat{i} + (10 - 6) \, \hat{j} = -2 \, \hat{i} + 4 \, \hat{j} \]

2. Now calculate \( \mathbf{a}_{avg} \):
\[ \mathbf{a}_{avg} = \frac{-2 \, \hat{i} + 4 \, \hat{j}}{4} = -0.5 \, \hat{i} + 1 \, \hat{j} \, \text{m/s}^2 \]

### (c) Average Speed
To find the average speed, we need the total distance traveled divided by the total time. However, we cannot determine the total distance from the given data since we only have the initial and final velocities and positions. The average speed is defined as:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\Delta t} \]
Since we cannot calculate the total distance traveled, we conclude that it is not possible to find the average speed from the given data.

### Summary of Results:
- **Average Velocity**: \( \mathbf{v}_{avg} = 1.25 \, \hat{i} + 0.5 \, \hat{j} \, \text{m/s} \)
- **Average Acceleration**: \( \mathbf{a}_{avg} = -0.5 \, \hat{i} + 1 \, \hat{j} \, \text{m/s}^2 \)
- **Average Speed**: Cannot be determined from the given data.