Question

A train is moving along a straight line with a constant acceleration a. A boy standing in the train throws a ball forward with a speed of 10m/s, with respect to the train at an angle of 600 to the horizontal. The boy has to move forward by 1.15m inside the train to catch the ball back at the initial height. The acceleration of the train is:
A. 5
B. 4
C. 6
D. 1

Answer 1

In this question, we will first use the required third equation of motion. As the ball has both vertical and horizontal motion, from this equation we will find the time and finally the acceleration of the train. This will give us the required result.
Formula used: $s = ut + \dfrac{1}{2}a{t^2}$

Complete answer:
Here, we will use the equation of motion. Since we know that the ball performs parabolic motion. We will find the acceleration of the train by using both vertical and horizontal motion of the ball.
For vertical motion, our equation can be written as:
\eqalign{ & s = ut - \dfrac{1}{2}g{t^2} \cr & \Rightarrow 0 = 10\sin {60^ \circ }t - \dfrac{1}{2} \times 10 \times {t^2} \cr & \therefore t = \sqrt {3s} \cr}
Now, we will study for the horizontal motion of the ball, which is given as:
$s = ut + \dfrac{1}{2}a{t^2}$
\eqalign{ & \Rightarrow 1.15 = 10\cos {60^ \circ } \times \sqrt 3 - \dfrac{1}{2}a(3) \cr & \therefore a = 5m/{s^2} \cr}

Therefore, we can say that option A) is correct, i.e., the acceleration of the train is given by the above result.

Additional information:
As we know that the equations of motion are equations which describe the behavior of a physical system in terms of its motion as a function of time. Further we can say that these equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. Here, dynamic variables are said to be normally spatial coordinates and time is used, but others are also possible, like momentum components and time.
Now, if we go in history, these equations of motion were discovered by Galileo Galilee but he could not manage to prove it practically that his equations were right or not. Later, Sir Isaac Newton proved these three equations of motion practically and also graphically. So, that is the reason now they are often called Newton’s three equations of motion. These equations tell us about the acceleration, displacement, time, final velocity of an object, initial velocity of an object.

Note:
Here we should remember that the three different equations of motion are used in finding different physical properties of a particle under motion. We should also observe that these equations are only applicable to the classical system not in the quantum system.