A car starts from rest and moves with uniform acceleration a on a straight road from time `t=0` to `t=T`. After that, a constant deceleration brings it to rest. In this process the average speed of the car is

Options

A. `(aT)/4`
B. `(3aT)/2`
C. `(aT)/2`
D. aT

Answer

To find the average speed of the car during its motion, we can break down the problem into two phases: the acceleration phase and the deceleration phase.

### Step-by-Step Solution:

1. **Acceleration Phase (from t = 0 to t = T):**
- The car starts from rest, so the initial velocity \( u = 0 \).
- The car accelerates uniformly with acceleration \( a \) for a time \( T \).
- The final velocity \( V \) at the end of this phase can be calculated using the formula:
\[ V = u + aT = 0 + aT = aT \]

2. **Distance Covered During Acceleration:**
- The distance \( s_1 \) covered during this acceleration can be calculated using the formula:
\[ s_1 = ut + \frac{1}{2} a t^2 = 0 + \frac{1}{2} a T^2 = \frac{1}{2} a T^2 \]

3. **Deceleration Phase (from t = T to t = 2T):**
- After time \( T \), the car starts decelerating to come to rest. The initial velocity for this phase is \( V = aT \).
- Let the deceleration be \( -b \). The car comes to rest, so the final velocity \( V_f = 0 \).
- Using the formula \( V_f = V - b(t - T) \), we can find the time taken to come to rest:
\[ 0 = aT - b(t - T) \implies b(t - T) = aT \implies t - T = \frac{aT}{b} \]
- Therefore, the total time for deceleration is \( t = T + \frac{aT}{b} \).

4. **Distance Covered During Deceleration:**
- The distance \( s_2 \) covered during deceleration can be calculated using:
\[ s_2 = Vt - \frac{1}{2} b(t - T)^2 \]
- Substituting \( V = aT \) and \( t - T = \frac{aT}{b} \):
\[ s_2 = aT \left( T + \frac{aT}{b} - T \right) - \frac{1}{2} b \left( \frac{aT}{b} \right)^2 \]
- Simplifying gives:
\[ s_2 = aT \cdot \frac{aT}{b} - \frac{1}{2} b \cdot \frac{a^2T^2}{b^2} = \frac{a^2T^2}{b} - \frac{1}{2} \frac{a^2T^2}{b} = \frac{1}{2} \frac{a^2T^2}{b} \]

5. **Total Distance and Total Time:**
- The total distance \( s \) is:
\[ s = s_1 + s_2 = \frac{1}{2} a T^2 + \frac{1}{2} \frac{a^2 T^2}{b} \]
- The total time \( t_{total} \) is:
\[ t_{total} = T + \frac{aT}{b} \]

6. **Average Speed Calculation:**
- The average speed \( v_{avg} \) is given by:
\[ v_{avg} = \frac{s}{t_{total}} = \frac{\frac{1}{2} a T^2 + \frac{1}{2} \frac{a^2 T^2}{b}}{T + \frac{aT}{b}} \]
- Simplifying this expression gives the average speed.

### Final Result:
The average speed of the car during the entire motion is:
\[ v_{avg} = \frac{aT}{2} \]