Question

An automobile and a truck start from rest at the same instant, with the automobile initially at some distance behind the truck. The truck has a constant acceleration of $2.2m/{s^2}$and the automobile has an acceleration of s$3.5m/{s^2}$. The automobile overtakes the truck when it (truck) has moved $60m$.
A) How much time does it take an automobile to overtake the truck?
B) How far was the automobile behind the truck initially?
C) What is the speed of each during overtaking?

Answer 1

Recall that when an object changes its position with respect to time it is said to be in motion. When the object moves in a straight line with a constant acceleration, it is represented by using the equation of motion. The relation between distance, initial velocity and acceleration is usually described by these equations of motion.

Step-By-Step solution:
Step I:
a) Let the initial velocity of the truck is zero$\Rightarrow u = 0$

Let the distance moved by the truck after $t$is

The distance covered by the truck is $s = 60m$

Acceleration is $a = 2.2m/{s^2}$

$s = ut + \dfrac{1}{2}a{t^2}$
$60 = 0 \times t + \dfrac{1}{2} \times 2.2 \times {t^2}$
${t^2} = \dfrac{{120}}{{2.2}}$
${t^2} = 54.54$
$t = 7.39\sec$

The time taken by the automobile to overtake the truck is$= 7.39\sec$

b) Let the distance between the automobile and truck be $= d$

The distance becomes $s = 60 + d$

The acceleration of the automobile is$= 3.5m/{s^2}$

Therefore the equation of motion becomes,

$d + 60 = 0 \times t + \dfrac{1}{2} \times 3.50 \times {t^2}$
$d + 60 = \dfrac{1}{2} \times 3.50 \times {(7.39)^2}$
$d + 60 = 95.57$
$d = 95.57 - 60$
$d = 35.57m$

The distance between the automobile and the truck is$= 35.57m$

c) Velocity of the truck will be

$v = u + at$
$v = 0 + 3.50 \times 7.39$
$v = 25.865m/s$

The velocity of truck is$= 25.865m/s$

Note: It is to be noted that the acceleration is defined as rate of change of velocity. If the acceleration of a body is changing at a faster rate then the speed of the body will also increase. Acceleration of a body has both magnitude and direction. So it is a vector quantity. Mass of an object does not change the acceleration, but if the force applied on the object is increased then its acceleration also increases. This means that the acceleration is directly proportional to the force applied on the body. But if the object is slowing down, it means that it is moving in an opposite direction to velocity and is said to be decelerating.