Question

A ball is thrown upward with an initial velocity of \(49 \mathrm{~m} / \mathrm{s}\) from \(539 \mathrm{~m}\) high. How high does the ball get, and how long does in take before it hits the ground? [Use results from the simple free fall problem, \(y^{\prime \prime}=-g\).]

Short answer

The ball will reach a maximum height of 589 meters, and it will take 31.8 seconds for it to hit the ground.

Step-by-step solution

Initial Setup

The ball is thrown from a height of 539 m with an initial velocity of 49 m/s. Let's let upwards be positive. Therefore, the acceleration due to gravity, \( g \), is -9.8 m/ \( s^2 \). Now we will plug into the position function: \( y(t) = y_0 + v_0*t - 0.5*g*t ^2 = 539 + 49t - 4.9t^2 \).

Calculate Maximum Height

To find the maximum height, we need to find when the ball's velocity is 0 (when it changes direction from moving upwards to falling). This is done by taking the derivative of y(t), which gives \( v(t) = y'(t) = v_0 - g*t = 49 - 9.8t \). Setting this equal to 0 gives \( t = v_0/g = 49/9.8 = 5 \) seconds. To find the maximum height, we substitute \( t = 5 \) back into the position function to get \( y(5) = 539 + 49*5 - 4.9*5^2 = 589 \) meters.

Calculate Total Time of Flight

To find out when the ball hits the ground, we set y(t) = 0. Solving \( 539 + 49t - 4.9t^2 = 0 \) gives two solutions \( t = 31.8 \) and \( t = -21.9 \). Since time cannot be negative, the total time of flight is \( 31.8 \) seconds.

Key concepts

Projectile Motion

Projectile motion is like the motion of a thrown ball, where the path is curved. It's a very common topic in physics. In projectile motion, an object moves in two dimensions and is influenced by gravity. One cool feature of projectile motion is that it includes both upward and downward movement.
Understanding projectile motion involves breaking down this motion into two separate components:

• Horizontal motion: This motion is constant because there's no horizontal acceleration in the absence of air resistance.
• Vertical motion: Here, gravity acts as a downward force, causing acceleration.

In the case of the ball, it started with an upward throw, so it followed a path that first went up and then came down. The highest point of the motion is what we call the maximum height. At this point, the vertical velocity temporarily becomes zero before the ball starts falling back down. This is a key moment in projectile motion.

• The path followed by the projectile is called a trajectory, which for a simple problem like this, takes the shape of a parabola.
• The time it takes to reach the highest point is key to calculating how high the ball goes before it starts its descent.

Free Fall

Free fall refers to the motion of an object where gravity is the only force acting upon it. It’s a fascinating part of physics because it shows how all objects fall at the same rate under the influence of gravity, regardless of their mass (ignoring air resistance).
In the ball scenario, when the ball was thrown upward, it was initially not in free fall. But once it reached its peak and began descending, it was effectively in free fall all the way to the ground.
During free fall:

• The only acceleration is due to gravity, which on Earth is approximately -9.8 m/s².
• Objects will accelerate uniformly, meaning their speed increases by 9.8 m/s every second in the downward direction.

As our exercise showed, the time from the peak to when the ball hit the ground was part of this free fall. By using the equation for vertical motion, it's possible to calculate how long it takes before an object hits the ground. Importantly, by setting up our equations correctly, we discovered the total flight time and the maximum height reached.

Calculus in Physics

Calculus is like the mathematical language of change and motion, which makes it super important in physics. Why? Because many aspects of physics, like motion, are all about things changing over time. Whether it's a car speeding up or a ball being thrown, calculus helps us understand what's happening, step by step.
In the problem of the ball, calculus is used to:

• Determine the velocity, which is the first derivative of the position function.
• Find the point in time when the velocity is zero, indicating the maximum height of the projectile.

Derivatives in calculus tell us a lot about the essence of motion. For example:

• By taking the derivative of the position function, you get velocity, telling you how speed changes over time.
• Taking a further derivative can give you acceleration, revealing how velocity is changing.

The calculations we did found the maximum height by setting the derivative (velocity) to zero. This process is a typical way calculus is applied in physics to solve motion-related problems. It gives us the tools to predict and understand movements easily and accurately.