Question

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. Calculate the average speed of the car.

Answer 1

The uniform acceleration is if an object's speed (velocity) is increasing at a constant rate then we say it has uniform acceleration. The rate of acceleration is constant. If a car speeds up then slows down then speeds up it doesn't have uniform acceleration. Divide the whole way into two paths $S_1$, $S_2$. Initial velocity on $S_1$ path is 0.

Step by step solution
Journey starts on t=0
Journey ends on t=T
According to average speed formula:-
Total distance/total time=$\dfrac{{S_1 + S_2}}{{T_1 + T_2}}$....... equation (1)
According to distance formula${S_1} = ut + \dfrac{1}{2}a{T^2}$.....[where u=0]
${S_1} = \dfrac{1}{2}a{T^2}$
According to the question rest of the path there is a deceleration
Let the deceleration V
$v = u + at$
$\Rightarrow v = 0 + at$ [where u=0]
$\Rightarrow$v=at........[value of initial velocity on starting point of $S_2$]...... equation (2)
Again for $S_2$ path:
$v = u + {a_2}{T_2}$
$\Rightarrow 0 = v + {a_2}{T_2}$ ${a_2}{T_2} = - aT$ .......[here end velocity v=0 and u=v]
${a_2}{T_2} = - aT$......... equation (3)
Now, find the $S_2$ distance

....[in $S_2$ the initial velocity in u=v]

{{S_2} = {\text{ }}aT{T_2} - \dfrac{1}{2}aT{T_2}{\text{ }}} \\\ {{S_2} = aT\left( {{T_2} - \dfrac{1}{2}{T_2}} \right)} \\\ {{S_2} = \dfrac{1}{2}{\text{ }}aT{T_2}{\text{ }}} \end{array}$$ [ putting the value of v=aT from equation (2) and $a_2T_2 = -aT$ from equation (3)] ........ equation (4) Now put the value of $S_1$, $S_2$ in equation 1 Average speed=$\dfrac{{\dfrac{1}{2}a{T^2} + \dfrac{1}{2}aT \times {T_1}}}{{T + {T_1}}}$ $ \Rightarrow \dfrac{{\dfrac{1}{2}aT(T + {T_1})}}{{T + {T_1}}} = \dfrac{1}{2}aT$ **Note** In this type of problem divide the whole pathe into two parts...and always the ending velocity of path1 is the initial velocity of path2. Now there is a question: what is average speed anyway? The average speed of an object is the total distance traveled by the object divided by the elapsed time to cover that distance. It's a scalar quantity which means it is defined only by magnitude. A related concept, average velocity, is a vector quantity.