Problem 16 - Basic Technical Mathematics with Calculus

Question

A stone is thrown straight up from the edge of a 45.0-m-high cliff. A loose stone at the edge of the cliff falls off 1.50 s later. What is the vertical velocity of the first stone, if the two stones reach the ground below at the same time?

Step-by-Step Solution

Answer

The initial vertical velocity of Stone A is approximately 14.3 m/s upwards.

Step 1: Understanding the Problem

We have two stones: Stone A is thrown upwards from a 45.0 m cliff, and Stone B falls off 1.50 seconds later. We need to find Stone A's initial vertical velocity, such that both stones hit the ground simultaneously.

Step 2: Use Known Equations of Motion

For both stones, we will use the kinematic equation: \[ y = y_0 + v_0t + \frac{1}{2} a t^2 \]where \(y\) is the final position, \(y_0\) is the initial position, \(v_0\) is the initial velocity, \(a\) is the acceleration (gravity, \(9.8\, \text{m/s}^2\)), and \(t\) is the time.

Step 3: Stone B's Time to Fall

Since Stone B falls freely from rest, it only has 1.50 seconds less than Stone A to reach the ground. Let \(t_A\) be the total time taken by Stone A to reach the ground, then Stone B takes \(t_B = t_A - 1.5\) seconds.

Step 4: Solve for Stone B's Time to Fall

Using \(y = 0\), \(y_0 = 45.0\, \text{m}\), \(v_0 = 0\), and \(a = 9.8\, \text{m/s}^2\), the equation for Stone B becomes:\[ 0 = 45.0 + 0 + \frac{1}{2} (9.8) (t_A - 1.5)^2 \]Solving this equation gives the falling time of Stone B.

Step 5: Solve for Stone A's Initial Velocity

For Stone A, the equation is:\[ 0 = 45.0 + v_0 t_A - \frac{1}{2} (9.8) t_A^2 \]With \(t_A\) from Step 4, solve this equation to find \(v_0\), the initial velocity of Stone A.

Key Concepts

Initial Velocity

When dealing with motion, initial velocity is a key starting point. It's the speed at which an object begins its journey. For our stone, the initial velocity (denoted as \(v_0\)) decides how high it goes and how long it stays in the air before retracing its path downwards.
Knowing the initial velocity helps determine its trajectory, specifically in how it competes against the forces pulling it back to earth.

• The stone is thrown upwards, so it carries a positive initial velocity.
• The velocity determines how quickly it reaches its peak height.
• A higher initial velocity would mean a longer ascent before gravity pulls it back.

In the given problem, finding the right initial velocity ensures our stone syncs timing perfectly with the second stone falling freely. This involves balancing against the stone's upward momentum and gravity's constant pull.

Acceleration Due to Gravity

The acceleration due to gravity is an unchanging force acting on the stone. Represented by \(g\) and measured as \(9.8 \text{ m/s}^2\), it pulls everything towards the Earth. This constant ensures that objects in free fall, like Stone B, accelerate downwards no matter their mass or the presence of an initial push.
Gravity affects both the ascent and the descent of Stone A.

• It slows down Stone A as it goes up, eventually bringing it to a stop before it falls back down.
• On the way down after the upward journey, gravity accelerates the stone back towards the ground.
• For Stone B, gravity is the only force at work, making it a pure example of free fall.

Understanding gravity's effect is crucial. It helps predict how long a thrown object stays in the air and how fast it falls back to earth. Its constant value simplifies calculations, allowing equations of motion to yield insights about timing and velocities.

Time of Flight

Time of flight describes the total duration an object spends in motion. This involves both the journey up and down for our stone. For Stone A, it's the key to landing at the same time as Stone B, which falls freely, but starts its fall 1.50 seconds later.
To solve the problem, one must calculate the time Stone A takes to ascend, stall, and then descend.

• The total time of flight for a projectile thrown upwards encompasses ascent, peak, and descent.
• For both stones to hit the ground together, their overall time intervals in flight must result in simultaneous landings.
• The kinematic equations, with insights from initial velocity and gravitational effects, are used to calculate and align these times.

By deriving expressions from these kinematic equations, we can ensure both stones, despite different starting conditions, reach the same end destination simultaneously. This highlights how carefully calibrated initial conditions, combined with physical constants like gravity, allow for precise predictions of motion.