Question

Particle A is projected vertically upward from a top of a tower. At the same time particle \(B\) is dropped from the same point. The graph of distance (s) between the two particle varies with time is.

Short answer

The distance (s) between particles A and B varies linearly with time (t), and can be represented by the equation \(s = ut\), where u is the initial velocity of particle A. The graph for this relation is a straight line with a positive slope (u) and has the distance s on the vertical axis and time t on the horizontal axis.

Step-by-step solution

Write the equations of motion for particles A and B

Since particle A is projected vertically upward, its initial velocity will be a non-zero value. We'll denote the initial velocity of A as \(u\). Particle B, on the other hand, is dropped from rest, meaning its initial velocity is equal to 0. Let's denote the acceleration due to gravity as \(g\) (a negative value). We can then write the equations of motion for particles A and B using the second equation of motion: For particle A: \(s_A = ut - \frac{1}{2}gt^2\) For particle B: \(s_B = -\frac{1}{2}gt^2\)

Find the distance between particles A and B

To find the distance \(s\) between the particles, we'll subtract particle B's displacement from particle A's displacement: \(s = s_A - s_B = (ut - \frac{1}{2}gt^2) - (-\frac{1}{2}gt^2) = ut - \frac{1}{2}gt^2 + \frac{1}{2}gt^2\) Simplifying the equation, we get: \(s = ut\)

Plot the graph of distance between particles A and B

The equation we derived in step 2 is a linear equation of the form \(s = ut\), where \(s\) is the distance between the particles, and \(t\) is the time. This means that the graph of distance between the particles A and B versus time will be a straight line with a positive slope (u). To plot the graph, we will have the distance s on the vertical axis and time t on the horizontal axis. At the initial state (t=0), the distance between the particles will be 0, which will be the starting point of the graph. As time progresses, the distance between the particles will increase linearly, and the graph will have a positive slope equal to the initial velocity of particle A (u).

Key concepts

Equations of Motion

In physics, the equations of motion are a set of formulas that describe how the position of an object changes over time under the influence of forces. These equations help us predict future positions and speeds of moving bodies.

For a vertically projected particle, like our particle A, the equation of motion can be expressed as:

• Particle A: \( s_A = ut - \frac{1}{2}gt^2 \)
• Particle B: \( s_B = -\frac{1}{2}gt^2 \)

These formulas take into account the initial velocity (\(u\)) and the acceleration due to gravity (\(g\)). Here, \(s_A\) and \(s_B\) represent the displacements of particles A and B, respectively. The symbol \(t\) stands for time.

The equations help determine how far an object moves in a specific time and the influence of gravity on its motion.

Vertical Projection

When a particle is projected vertically upward, as with Particle A in our problem, it starts with an initial velocity directed opposite to the force of gravity.

The motion can be described by the equation \( s_A = ut - \frac{1}{2}gt^2 \), where \(u\) is the initial velocity. This is different from an object that is simply dropped, like Particle B, which starts its journey at zero velocity.

The unique aspect of vertical projection is that it has both ascent and descent phases. In the ascending phase, the velocity of the particle decreases until it reaches the peak point, where its velocity becomes zero. Thereafter, it descends back towards the ground.

Vertical projection is a fascinating study as it beautifully illustrates the interplay between an object's initial motion and gravitational forces.

Acceleration due to Gravity

Acceleration due to gravity is a fundamental physical constant denoted by \(g\), typically valued at approximately \(9.81 \text{ m/s}^2 \) on the surface of Earth. It represents the rate at which an object speeds up as it falls freely towards the Earth.

In equations of motion, \(g\) is often considered negative when analyzing vertical upward motion, as gravity acts in the opposite direction to the object's initial motion.

Understanding acceleration due to gravity is crucial because it influences how quickly an object's velocity changes with time. Every object, irrespective of its mass, experiences the same gravitational acceleration when free-falling, assuming air resistance is negligible.

Therefore, gravity plays a vital role in shaping the trajectory of objects, influencing both their ascent and descent.

Displacement

In physics, displacement refers to the change in position of an object. It's a vector quantity, which means it has both magnitude and direction.

For our particular problem, displacement is described for two particles, A and B, with their specific equations of motion. The difference between their displacements gives us the distance \(s\) between them. This is calculated as:

• \( s = s_A - s_B = ut \)

This simplifies to a linear relation where distance is proportional to time. This linearity results from the uniform initial velocity \(u\) with no further horizontal acceleration.

Displacement can be different from distance when considering direction, but in this scenario, we only look at the magnitude of movement relative to their starting point.

Graphical Representation

Graphical representation of motion helps visualize how distance varies with time. For our problem involving particles A and B, this is depicted as a linear graph.

The equation \( s = ut \) informs us that distance increases linearly with time. The graph of distance \((s)\) versus time \((t)\) is a straight line starting at the origin \((0,0)\), indicating that initially, there's no distance between the particles. As time progresses, the line's positive slope reflects the constant velocity, \(u\), of particle A.

Key aspects to note in graphs of motion:

• Slope indicates velocity.
• Straight lines suggest uniform velocity.
• The position where the line meets the time axis gives initial conditions.

Employing graphical representation unlocks a visual understanding of an object's dynamics, enriching our comprehension of motion's complexities.