If acceleration of a particle at any time is given by `a = 2t + 5`
Calculate the velocity after 5 s, if it starts from rest

Options

A. `50 ms ^(-1)`
B. `25 ms ^(-1)`
C. `100 ms ^(-1)`
D. `75ms^(-1)`

Answer

To solve the problem of finding the velocity of a particle after 5 seconds given its acceleration as \( a = 2t + 5 \) and that it starts from rest, we can follow these steps:

### Step 1: Understand the relationship between acceleration, velocity, and time
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as:
\[ a = \frac{dv}{dt} \]
Given \( a = 2t + 5 \), we can write:
\[ \frac{dv}{dt} = 2t + 5 \]

### Step 2: Rearrange the equation for integration
We can rearrange the equation to separate the variables:
\[ dv = (2t + 5) dt \]

### Step 3: Integrate both sides
Now we will integrate both sides. The left side will be integrated with respect to \( v \) and the right side with respect to \( t \):
\[ \int dv = \int (2t + 5) dt \]
The limits for \( v \) will be from 0 (initial velocity) to \( v \) (final velocity), and the limits for \( t \) will be from 0 to 5 seconds.

### Step 4: Perform the integration
Integrating the left side gives:
\[ v = \int (2t + 5) dt \]
Now we integrate the right side:
\[ \int (2t + 5) dt = \int 2t dt + \int 5 dt = t^2 + 5t + C \]
where \( C \) is the constant of integration.

### Step 5: Apply the limits of integration
Now we apply the limits from 0 to 5:
\[ v = \left[ t^2 + 5t \right]_0^5 \]
Calculating this:
\[ v = (5^2 + 5 \cdot 5) - (0^2 + 5 \cdot 0) = (25 + 25) - 0 = 50 \]

### Step 6: Conclusion
Thus, the velocity of the particle after 5 seconds is:
\[ v = 50 \, \text{m/s} \]

### Summary
The final velocity after 5 seconds, starting from rest, is \( 50 \, \text{m/s} \).

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