Question

A ball is thrown directly downward, with an initial speed of \(10.0 \mathrm{~m} / \mathrm{s}\), from a height of \(50.0 \mathrm{~m}\). After what time interval does the ball strike the ground?

Short answer

Answer: Approximately 1.783 seconds.

Step-by-step solution

Identify the given values and constants

We are given the initial speed (\(v_0 = 10.0 \mathrm{~m/s}\)) and the initial height (\(h_0 = 50.0 \mathrm{~m}\)). Since the ball is thrown downward, the acceleration due to gravity (\(g = 9.81 \mathrm{~m/s^2}\)) will act in the same direction as the initial velocity.

Choose the appropriate kinematic equation

We will use the following kinematic equation, which relates displacement, initial velocity, time, and acceleration: $$h - h_0 = v_0t + \frac{1}{2}gt^2$$ Since the initial height is \(h_0\) and the ball is falling down, the final height, \(h\), will be 0. Plugging in the given values, we have: $$0 - 50.0 = 10.0t + \frac{1}{2}(9.81)t^2$$

Rearrange the equation and solve for the time

We can rearrange the above equation to solve for time, \(t\): $$\frac{1}{2}(9.81)t^2 + 10.0t - 50.0 = 0$$ This is a quadratic equation in the form of \(at^2 + bt + c = 0\). We can use the quadratic formula to solve for \(t\): $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, \(a = \frac{1}{2}(9.81)\), \(b = 10.0\), and \(c = -50.0\). Plugging the values, we get: $$t = \frac{-10.0 \pm \sqrt{(10.0)^2 - 4(\frac{1}{2}(9.81))(-50.0)}}{2(\frac{1}{2}(9.81))}$$ Simplifying and solving for \(t\), we get two solutions: \(t \approx 1.783\,\mathrm{s}\) and \(t \approx -5.668\,\mathrm{s}\).

Choose the appropriate solution for time

Since it is not possible for the time interval to be negative, we discard the negative solution and consider the positive one: \(t \approx 1.783\,\mathrm{s}\) After approximately \(1.783\) seconds, the ball will strike the ground.

Key concepts

Free Fall Motion

When an object is dropping solely under the influence of gravity, it's said to be in free fall motion. Objects in free fall acceleration move faster as time progresses because of the constant force of gravity pulling them towards the Earth. It's essential to understand that the term free fall does not consider air resistance – the objects are assumed to move in a vacuum.

For example, if you dropped a ball straight down, it would keep accelerating at a constant rate until it hits the ground. This acceleration is described by the acceleration due to gravity, which is approximately 9.81 meters per second squared (\(9.81\text{ m/s}^2\text{)\) on Earth. The kinematic equations for free fall motion allow us to predict how long it will take for the ball to hit the ground and what velocity it will have at any point in its fall.

It's vital to distinguish between dropping (where initial velocity is 0) and throwing an object downwards (where there's an initial velocity). In our exercise, the ball is thrown directly downward, which means it already has a speed when the motion starts, adding complexity to the calculations.

Quadratic Formula

The quadratic formula is a fundamental tool in algebra for finding the solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions, or roots, are given by the formula:

\[x = \frac{-b \[5px] \pm \sqrt{b^2 - 4ac}}{2a}\]
When applied to kinematic equations, this formula can determine specific points in time, like when an object will reach the ground during free fall. The quadratic equation has a maximum of two real solutions, which represent two possible times at which the object could reach a particular position.

In the context of our exercise, a positive root indicates the actual time it will take the thrown ball to hit the ground. On the other hand, a negative root doesn't have a real-world application in this scenario – you can't go back in time, so we only take the positive value as the solution.

Acceleration Due to Gravity

Gravity is a force that attracts two bodies towards each other, and on Earth, it gives objects an acceleration due to gravity of approximately \(9.81\text{ m/s}^2\), denoted as \(g\). This acceleration is why objects, when dropped, will fall towards the ground at increasing speeds.

The influence of gravity on objects in free fall is always in the direction towards the center of the Earth; hence it's considered a negative value when calculating the motion of an object thrown upwards. However, when an object is thrown downwards, gravity acts in the same direction as the object's movement, thus we consider it a positive value.

In our ball-throwing scenario, gravity is accelerating the ball in the same direction as its initial velocity, making the ball hit the ground sooner than if it were just dropped.

Initial Velocity

The initial velocity is the velocity that an object has when it starts its motion. This value is essential when calculating the future positions or velocities of the object using kinematic equations. In the context of free fall, the initial velocity dictates how the object's motion will progress under the influence of gravity.

If, for instance, an object is dropped, its initial velocity is 0. But if the object is thrown downwards, like the ball in our exercise, it starts with an initial velocity in the same direction as gravity, resulting in a shorter time to hit the ground compared to an object that was simply dropped. This is because the ball already has a head start in speed that continues to increase due to gravity's acceleration.

The initial velocity can be positive, negative, or zero. In our example, the positive initial velocity represents the ball's initial speed directed downward, accelerating its descent towards the ground.