The initial position of an object at rest is given by 3i - 8j. It moves with constant acceleration and reaches the position 2i + 4j after 4s. What will be its acceleration?

To solve this problem, we need to determine the acceleration of the object using the information about its initial and final positions, and the time it took to reach the final position.
Given:
• Initial position: r⃗i=3i^−8j^\vec{r}_i = 3\hat{i} - 8\hat{j}
• Final position after 4 seconds: r⃗f=2i^+4j^\vec{r}_f = 2\hat{i} + 4\hat{j}
• Time taken: t=4 secondst = 4 \, \text{seconds}
• The object is initially at rest, so the initial velocity v⃗i=0\vec{v}_i = 0.
Step 1: Use the equation of motion
The general equation of motion for an object moving with constant acceleration is:
r⃗f=r⃗i+v⃗it+12a⃗t2\vec{r}_f = \vec{r}_i + \vec{v}_i t + \frac{1}{2} \vec{a} t^2
Where:
• r⃗f\vec{r}_f is the final position,
• r⃗i\vec{r}_i is the initial position,
• v⃗i\vec{v}_i is the initial velocity,
• a⃗\vec{a} is the acceleration,
• tt is the time.
Step 2: Substitute the known values
Since the object starts from rest, v⃗i=0\vec{v}_i = 0. The equation simplifies to:
r⃗f=r⃗i+12a⃗t2\vec{r}_f = \vec{r}_i + \frac{1}{2} \vec{a} t^2
Now, substitute the values for r⃗f\vec{r}_f, r⃗i\vec{r}_i, and tt:
2i^+4j^=3i^−8j^+12a⃗(4)22\hat{i} + 4\hat{j} = 3\hat{i} - 8\hat{j} + \frac{1}{2} \vec{a} (4)^2
Simplify:
2i^+4j^=3i^−8j^+8a⃗2\hat{i} + 4\hat{j} = 3\hat{i} - 8\hat{j} + 8 \vec{a}
Step 3: Rearrange the equation
Rearranging the terms:
(2i^+4j^)−(3i^−8j^)=8a⃗(2\hat{i} + 4\hat{j}) - (3\hat{i} - 8\hat{j}) = 8 \vec{a}
Simplifying the left side:
−i^+12j^=8a⃗- \hat{i} + 12 \hat{j} = 8 \vec{a}
Step 4: Solve for acceleration a⃗\vec{a}
Now, divide both sides by 8 to solve for a⃗\vec{a}:
a⃗=−i^+12j^8\vec{a} = \frac{-\hat{i} + 12\hat{j}}{8} a⃗=−18i^+128j^\vec{a} = -\frac{1}{8}\hat{i} + \frac{12}{8}\hat{j} a⃗=−0.125i^+1.5j^\vec{a} = -0.125\hat{i} + 1.5\hat{j}
Conclusion:
The acceleration of the object is:
a⃗=−0.125i^+1.5j^ m/s2\boxed{\vec{a} = -0.125 \hat{i} + 1.5 \hat{j} \, \text{m/s}^2}
This means the acceleration has a component of −0.125 m/s2-0.125 \, \text{m/s}^2 in the i^\hat{i} direction (along the x-axis) and a component of 1.5 m/s21.5 \, \text{m/s}^2 in the j^\hat{j} direction (along the y-axis).