Problem 104 - Fundamentals of Physics

Question

A ball is thrown horizontally from a height of \(20 \mathrm{~m}\) and hits the ground with a speed that is three times its initial speed. What is the initial speed?

Step-by-Step Solution

Answer

The initial speed of the ball is approximately 7 m/s.

Step 1: Understand the problem

The ball is thrown horizontally from a height of 20 m. It hits the ground with a speed that is three times its initial speed. We need to find the initial horizontal speed of the ball.

Step 2: Analyze vertical motion

The vertical motion is due to gravity, which affects only the vertical component of the velocity. The ball falls 20 m to the ground. We use the formula for free fall distance:\[ h = \frac{1}{2} g t^2 \]where \( h = 20 \ \rm m \), and \( g = 9.8 \ \rm m/s^2 \). We solve for \( t \), the time of flight.

Step 3: Solve for time of flight

Rearrange the equation for \( t \):\[ 20 = \frac{1}{2} \cdot 9.8 \cdot t^2 \]\[ 40 = 9.8 t^2 \]\[ t^2 = \frac{40}{9.8} \]\[ t = \sqrt{\frac{40}{9.8}} \approx 2.02 \ \rm s \]

Step 4: Use horizontal motion

The horizontal velocity component \( v_{0} \) is constant. The total horizontal displacement or launch speed does not change in the horizontal direction. When the ball hits the ground, its speed is given as 3 times its initial horizontal speed. This implies horizontal speed when it hits the ground is \( 3v_{0} \).

Step 5: Consider horizontal velocity component at impact

At the moment of impact, the horizontal component of the velocity remains \( v_{0} \). The vertical component has increased due to gravity. Use Pythagoras' theorem to relate the horizontal and vertical components of velocity at impact:\[ v = \sqrt{v_{0}^2 + (gt)^2} \]where \( v = 3v_{0} \).

Step 6: Solve for initial speed

Plug \( v = 3v_{0} \) into the equation:\[ 3v_{0} = \sqrt{v_{0}^2 + (9.8 \cdot 2.02)^2} \]\[ 9v_{0}^2 = v_{0}^2 + (9.8 \cdot 2.02)^2 \]\[ 9v_{0}^2 - v_{0}^2 = (9.8 \cdot 2.02)^2 \]\[ 8v_{0}^2 = 392.04 \]\[ v_{0}^2 = \frac{392.04}{8} \approx 49.005 \]\[ v_{0} = \sqrt{49.005} \approx 7 \ \rm m/s \]

Step 7: Conclusion: Initial speed

The initial horizontal speed of the ball is approximately \(7 \ \rm m/s\).

Key Concepts

Horizontal and Vertical Components of Velocity

When dealing with projectile motion, it's vital to separate the horizontal and vertical components of velocity. This makes calculations simpler and more intuitive. Imagine a ball is thrown. Its path is not a straight line but forms a curve. This happens because both horizontal displacement and vertical displacement occur simultaneously.

The horizontal component of velocity, often denoted as \(v_0\), remains constant as there is no acceleration in the horizontal direction (ignoring air resistance). Therefore, it does not change during the entire flight of the projectile. For our example, the initial horizontal speed was the factor of interest.

In contrast, the vertical component of velocity is affected by gravity. This means it will change as time passes. Initially, if a ball is thrown horizontally, its vertical velocity component is zero. But as gravity acts, the ball accelerates downward.

• Horizontal Velocity: Constant
• Vertical Velocity: Changes due to gravity

By focusing separately on these components, we can easily compute the overall speed at any point using Pythagoras' Theorem, combining both dimensions.

Kinematics Equations

Kinematics equations serve as fundamental tools when analyzing projectile motion. They allow us to describe motion in terms of displacement, velocity, and acceleration.

For vertical motion under the influence of gravity (like when a ball is falling), the kinematic equation used was:\[ h = \frac{1}{2} g t^2 \]Here, \( h \) is the height, \( g \) represents acceleration due to gravity, and \( t \) is time. This equation provides a way to find the time it takes for the ball to fall a specific height.

• This formula is specifically for freely falling objects where the initial vertical velocity is zero.
• It assumes no other forces are at work except gravity, like air resistance.

Using the time calculated from the vertical movement to analyze horizontal movement can help determine aspects like initial speed. Additionally, Pythagoras' Theorem integrates these concepts to provide a complete picture of impact speed, taking into account resultant speed derived from both horizontal and vertical components.

Free Fall

Free fall describes an object's motion where gravity is the sole force acting on it. In this state, the object experiences acceleration directed vertically downward.

In the ball-throwing scenario, while the ball experiences horizontal motion, its vertical motion is a classic example of free fall. Initially, when the ball is thrown horizontally, there is no vertical speed. However, as time progresses, gravity causes the ball to accelerate downward, increasing its vertical velocity.

• With free fall, regardless of horizontal motion, the object accelerates at \(9.8 \, \text{m/s}^2\) downward.
• This uniform acceleration causes the ball to gain speed as it descends, impacting its final speed when it hits the ground.

By calculating the vertical velocity at the point of impact, using the elapsed time from the kinematics equations, we can gain a clearer understanding of the overall speed, since it combines with the steady horizontal speed to contribute to the impact velocity profile.