a) 12 A particle moves in a straight line. Its velocity became \( 100 \mathrm{~cm} / \mathrm{sec} \). after 5 seconds, and became \( 72 \mathrm{~cm} / \mathrm{sec} \). in the same direction after 12 seconds from starting motion, then its initial velocity \( =\ldots \ldots \mathrm{cm} / \mathrm{sec} \) \( \begin{array}{llll}\text { a) }-4 & \text { b) } 86 & \text { c) } 120 & \text { d) } 172\end{array} \)

To find the initial velocity of the particle, we can use the average acceleration formula. The formula for acceleration \( a \) is given by: \[ a = \frac{v_f - v_i}{t} \] Where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is the time. From the question, we are given that the velocity changes from \( v_i \) to \( 100 \, cm/sec \) in 5 seconds and from \( 100 \, cm/sec \) to \( 72 \, cm/sec \) in the next 7 seconds (the total time is 12 seconds). Let’s consider the time intervals. 1. From \( t = 0 \) to \( t = 5 \): - \( v_f = 100 \, cm/s \) - \( t = 5 \) - We need to find \( v_i \). 2. From \( t = 5 \) to \( t = 12 \): - \( v_i = 100 \, cm/s \) - \( v_f = 72 \, cm/s \) - \( t = 7 \) Now let's calculate the acceleration during the second interval: \[ a = \frac{72 - 100}{7} = \frac{-28}{7} = -4 \, cm/s^2 \] Now we can find the initial velocity. In the first 5 seconds, we can also apply the same formula. The acceleration can be assumed to remain constant (which is a common approach unless stated otherwise), hence we can calculate the initial velocity. Set up the equation: \[ 100 = v_i + (-4)(5) \] \[ 100 = v_i - 20 \] \[ v_i = 120 \, cm/s \] Thus, the initial velocity is \( \boxed{120 \, cm/s} \), which corresponds to option c. Now, let’s add some interesting information related to motion that might evoke curiosity! To understand the concept of motion better, it helps to dive into the history of classical mechanics, where figures like Galileo first studied the principles of motion in the 16th century. His experiments with inclined planes laid groundwork that would evolve into Newton's laws of motion, defining not just speed but also acceleration and force, the building blocks of classical physics today! For those wanting to delve deeper, Jean-Pierre Maury's *Mechanics for Beginners* can be an insightful read, providing an engaging foundation in kinematics and dynamics. And if you're particularly intrigued by how these principles apply to real-world scenarios, checking out how engineers use these calculations when designing rollercoasters could transform your understanding of speed and thrills!