Problem 48 - Master Resource Book in JEE Main Physics

Question

A stone is allowed to fall from the top of a tower \(100 \mathrm{~m}\) high and at the same time another stone is projected vertically upwards from the ground with a velocity of \(254 \mathrm{~ms}^{-1}\). The two stones will meet after \(\begin{array}{llll}\text { (a) } 4 \mathrm{~s} & \text { (b) } 0.4 \mathrm{~s} & \text { (c) } 0.04 \mathrm{~s} & \text { (d) } 40 \mathrm{~s}\end{array}\)

Step-by-Step Solution

Answer

The stones meet after approximately 0.4 seconds, which is option (b).

Step 1: Identify the variables and equations

Let's denote the height of the tower as \( h = 100 \ m \). The acceleration due to gravity is \( g = 9.8 \ m/s^2 \). For the first stone released from the top, its initial velocity \( u_1 = 0 \). For the second stone projected upwards, initial velocity \( u_2 = 254 \ m/s \). We need to find the time \( t \) when they meet somewhere between the top and bottom of the tower.

Step 2: Equation for downward stone

For the stone falling from the top, we use the equation for distance \( s = u_1t + \frac{1}{2}gt^2 \). Since \( u_1 = 0 \), the distance \( s_1 \) that this stone travels is.\[s_1 = \frac{1}{2}gt^2 = \frac{1}{2} \cdot 9.8 \cdot t^2 = 4.9t^2\]

Step 3: Equation for upward stone

For the stone projected upwards, we use the equation \( s = ut - \frac{1}{2}gt^2 \) (since gravity acts downward and retards the stone). The distance \( s_2 \) it travels is:\[s_2 = 254t - \frac{1}{2} \cdot 9.8 \cdot t^2 = 254t - 4.9t^2\]

Step 4: Setting equations for meeting point

The two stones meet when the sum of distances they travel is equal to the height of the tower (100 m):\[s_1 + s_2 = 100\]Substitute the expressions for \( s_1 \) and \( s_2 \):\[4.9t^2 + (254t - 4.9t^2) = 100\]

Step 5: Simplify and solve

Simplify the equation:\[254t - 100 = 0\]Solve for \( t \):\[254t = 100\]\[t = \frac{100}{254}\]\[t \approx 0.3945 \, s\]

Step 6: Choose closest option

The closest option to the calculated time \( t \approx 0.3945 \, s \) is option (b) \( 0.4 \, s \).

Key Concepts

Equation of motion

The equations of motion are essential tools in the study of kinematics. They help us predict and describe the motion of objects in detail. These equations are particularly useful when we know some parameters of an object's motion, such as initial velocity, final velocity, acceleration, and time—and we wish to find another unknown parameter.

In this exercise, we used two specific equations of motion:

• For the stone falling from the tower, its equation is given by: \[ s = u_1t + \frac{1}{2}gt^2 \]
• For the stone projected upwards, the equation becomes: \[ s = ut - \frac{1}{2}gt^2 \]

These formulas allow us to calculate the distances traveled by both stones. By setting these two distances equal to the height of the tower, we can solve for the meeting time of the stones.

Projectile motion

Projectile motion describes the motion of an object thrown into the air, subject to only the acceleration due to gravity. This motion occurs in two dimensions and can be better understood by analyzing two components: horizontal and vertical.

In the given problem, we focus on vertical motion as one stone is projected upwards into the air and another falls down. Both experience gravity's pull, but one moves against it initially, and the other moves with it. The initial velocities, directions, and accelerations are all crucial to determine the stones' paths and meeting point.

• Upward motion follows: \[ s = ut - \frac{1}{2}gt^2 \]
• Downward motion follows: \[ s = \frac{1}{2}gt^2 \]

By considering these equations, we break down the movements and predict precisely where and when the two objects will intersect.

Free fall

Free fall refers to the motion of an object where gravity is the only force acting upon it. In this scenario, the falling stone from the tower experiences free fall. It begins with zero initial velocity and descends due to gravitational acceleration.

The critical attribute of free fall is the constant acceleration, typically represented by the symbol \(g\), which on Earth's surface is approximately \(9.8 \, m/s^2\).

• Initially, \(u = 0\)
• Distance covered is: \[ s = \frac{1}{2}gt^2 \]

This concept emphasizes that the object's position changes at an increasing rate, expressed through the quadratic term \(t^2\). Understanding this principle is vital for calculating when two objects, one experiencing free fall and the other projecting, will meet.

Acceleration due to gravity

Acceleration due to gravity, often denoted by \(g\), is a fundamental concept influencing every object near Earth's surface. It induces a constant acceleration of approximately \(9.8 \, m/s^2\) downward.

In the exercise, this consistent force affects both the stone falling from the tower and the stone projected upwards. Gravity accelerates the downward stone and simultaneously decelerates the upward-moving stone, eventually pulling it back down.

• The role of gravity in upward motion: \[ s = ut - \frac{1}{2}gt^2 \]
• The role of gravity in downward motion: \[ s = \frac{1}{2}gt^2 \]

Recognizing the uniform influence of \(g\) allows us to apply it in equations of motion to predict movement accurately even when projectiles move in opposite directions.