A particle moves in x-y plane according to equations `x = 4t^(2)+5t` and 6y=5t The acceleration of the particle must be

Options

A. `8m//sec^(2)`
B. `12 m//sec^(2)`
C. `14 m//sec^(2)`
D. none of the above

Answer

To find the acceleration of the particle moving in the x-y plane according to the given equations \( x = 4t^2 + 5t \) and \( 6y = 5t \), we can follow these steps:

### Step 1: Differentiate the position function for x to find the velocity in the x-direction.
The position function for x is given by:
\[ x(t) = 4t^2 + 5t \]

To find the velocity in the x-direction (\( v_x \)), we differentiate \( x \) with respect to time \( t \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(4t^2 + 5t) \]

### Step 2: Calculate the derivative.
Using the power rule:
\[ v_x = 8t + 5 \]

### Step 3: Differentiate the velocity function to find the acceleration in the x-direction.
Now, we differentiate \( v_x \) with respect to time \( t \) to find the acceleration in the x-direction (\( a_x \)):
\[ a_x = \frac{dv_x}{dt} = \frac{d}{dt}(8t + 5) \]

### Step 4: Calculate the derivative.
Differentiating gives:
\[ a_x = 8 \]
(The constant term \( 5 \) differentiates to \( 0 \)).

### Step 5: Find the position function for y and differentiate to find the velocity in the y-direction.
The position function for y is given by:
\[ 6y = 5t \]
To express \( y \) in terms of \( t \):
\[ y = \frac{5t}{6} \]

Now, differentiate \( y \) with respect to \( t \) to find the velocity in the y-direction (\( v_y \)):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}\left(\frac{5t}{6}\right) \]

### Step 6: Calculate the derivative.
Differentiating gives:
\[ v_y = \frac{5}{6} \]

### Step 7: Differentiate the velocity function to find the acceleration in the y-direction.
Now, we differentiate \( v_y \) with respect to time \( t \) to find the acceleration in the y-direction (\( a_y \)):
\[ a_y = \frac{dv_y}{dt} = \frac{d}{dt}\left(\frac{5}{6}\right) \]

### Step 8: Calculate the derivative.
Since \( \frac{5}{6} \) is a constant, its derivative is:
\[ a_y = 0 \]

### Step 9: Calculate the net acceleration of the particle.
The net acceleration \( a \) can be found using the formula:
\[ a = \sqrt{a_x^2 + a_y^2} \]

Substituting the values we found:
\[ a = \sqrt{(8)^2 + (0)^2} = \sqrt{64} = 8 \, \text{m/s}^2 \]

### Final Answer:
The acceleration of the particle is \( 8 \, \text{m/s}^2 \).

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