Question

A blimp is ascending at the rate of \(7.50 \mathrm{~m} / \mathrm{s}\) at a height of \(80.0 \mathrm{~m}\) above the ground when a package is thrown from its cockpit horizontally with a speed of \(4.70 \mathrm{~m} / \mathrm{s}\). a) How long does it take for the package to reach the ground? b) With what velocity (magnitude and direction) does it hit the ground?

Short answer

In summary, the package takes approximately 5.37 seconds to reach the ground and has a velocity of 45.4 m/s in a direction 84.4 degrees below the horizontal when it hits the ground.

Step-by-step solution

Part A: Find the time for the package to reach the ground

To find the time it takes for the package to reach the ground, we need to focus on the package's vertical motion. We know the initial vertical velocity is 7.50 m/s (the ascending rate of the blimp) and the initial vertical position is 80.0 m above the ground. In this case, we can use the kinematic equation: \(h = h_0 + v_{0y}t - \frac{1}{2}gt^2\) Here, \(h\) is the final height (0 m), \(h_0\) is the initial height (80.0 m), \(v_{0y}\) is the initial vertical velocity (7.50 m/s), \(t\) is the time, and \(g\) is the acceleration due to gravity (9.81 m/s²). Solving for \(t\), we have: \(0 = 80.0 + 7.50t - \frac{1}{2}(9.81)t^2\) Now, we can use the quadratic formula to find the time: \(t = \frac{-v_{0y} \pm \sqrt{v_{0y}^2 - 4(-\frac{1}{2}g)(h_0)}}{g}\) Plugging in the given values: \(t = \frac{-7.50 \pm \sqrt{7.50^2 - 4(-\frac{1}{2}(9.81))(80.0)}}{9.81}\) There will be two possible solutions for \(t\), but since we know the package is being thrown upward, we can ignore the negative value and choose the positive one: \(t \approx 5.37\) s So, it takes about 5.37 seconds for the package to reach the ground.

Part B: Determine the magnitude and direction of the velocity when the package hits the ground

In this part, we want to find the horizontal and vertical components of the package's final velocity right before it hits the ground. We can use the following equations to find the components: For the horizontal component \(v_x\): \(v_x = v_{0x}\) Here, \(v_{0x}\) is the initial horizontal velocity (4.70 m/s). The horizontal component of the velocity remains constant as there is no horizontal acceleration. \(v_x = 4.70\) m/s For the vertical component \(v_y\): \(v_y = v_{0y} - gt\) Here, \(v_{0y}\) is the initial vertical velocity (7.50 m/s), \(g\) is the acceleration due to gravity (9.81 m/s²), and \(t\) is the time (5.37 s). \(v_y = 7.50 - (9.81)(5.37) \approx -45.3\) m/s To find the magnitude of the final velocity \(v_f\), we can use the Pythagorean theorem: \(v_f = \sqrt{v_x^2 + v_y^2} = \sqrt{(4.70)^2 + (-45.3)^2} \approx 45.4\) m/s Finally, to find the direction of the final velocity, we can calculate the angle \(\theta\) below the horizontal using the arctangent function: \(\theta = \arctan{\frac{v_y}{v_x}} = \arctan{\frac{-45.3}{4.70}} \approx -84.4^\circ\) So, the package hits the ground with a velocity of about 45.4 m/s at an angle of 84.4 degrees below the horizontal.

Key concepts

Projectile Motion

Understanding projectile motion is essential to solving problems like the exercise involving a package thrown from a blimp. Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity and, if applicable, air resistance. However, for most introductory physics problems, air resistance is neglected, making things simpler.

There are two crucial points to remember about projectile motion:

• The horizontal motion of a projectile is constant because there are no forces acting in the horizontal direction (ignoring air resistance).
• The vertical motion is influenced by gravity, causing a constant downwards acceleration of -9.81 m/s². This results in a parabolic trajectory when graphed.

To solve projectile motion problems, we often break the motion into horizontal and vertical components and analyze each using kinematic equations.

Free Fall

The concept of free fall comes into play when the package is dropped from the blimp. Free fall is a specific type of projectile motion where the only force acting on the object is gravity. This means that, if we disregard air resistance, all objects in free fall near the Earth's surface accelerate downwards at the same rate, regardless of their mass.

The vertical motion of the package can be understood in terms of free fall once released. Despite its initial upward velocity, gravity will slow the package to a stop and then accelerate it back towards the ground. The kinematic equations are used to determine the time it takes for this motion to occur and the velocity at any point during the descent.

Quadratic Formula

The quadratic formula is a powerful mathematical tool to solve quadratic equations, which appear frequently in physics problems, including those on projectile motion. A quadratic equation is any equation of the form:
\[ ax^2 + bx + c = 0 \]
where, in this context, `x` would represent time, and `a`, `b`, and `c` are constants. The quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
provides the solution(s) for `x`, which can correspond to different points in time of the motion's trajectory. In the problem regarding the blimp, this formula was used to find out how long it took for the package to hit the ground.

Velocity Components

Understanding velocity components is crucial when dealing with projectile motion. Since motion occurs in a two-dimensional plane, we can describe the velocity of an object in terms of its horizontal component (\(v_x\)) and its vertical component (\(v_y\)).

In our exercise, velocity components explained the final motion of the package as it hit the ground:

• The horizontal velocity component remained constant as there is no horizontal acceleration (ignoring air resistance).
• The vertical velocity component changed due to the acceleration of gravity.

The final velocity of the package was found by combining these components, resulting in the package's speed and angle of impact upon the ground.