Question

An air-hockey puck has a model rocket rigidly attached to it. The puck is pushed from one corner along the long side of the \(2.00-\mathrm{m}\) long air- hockey table, with the rocket pointing along the short side of the table, and at the same time the rocket is fired. If the rocket thrust imparts an acceleration of \(2.00 \mathrm{~m} / \mathrm{s}^{2}\) to the puck, and the table is \(1.00 \mathrm{~m}\) wide, with what minimum initial velocity should the puck be pushed to make it to the opposite short side of the table without bouncing off either long side of the table? Draw the trajectory of the puck for three initial velocities: \(vv_{\min } .\) Neglect friction and air resistance.

Short answer

Answer: 0.707 m/s

Step-by-step solution

Identify the variables

In this problem, we have the following given information: - Length of air-hockey table (L) = 2.00 m - Width of air-hockey table (W) = 1.00 m - Acceleration due to the rocket thrust (a) = 2.00 m/s² - Minimum initial velocity along the x-axis (v_min) = ? We need to find the minimum initial velocity, v_min.

Analyze motion in the x and y directions

As the puck will be moving along both the length and width of the table, we can separate the motion into two components: x-direction (length) and y-direction (width). a_x = a (since the rocket is pointing along the short side) Then we will analyze the kinematic equations for both x and y directions, using the following equation: v_f = v_i + a*t where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and t is the time.

Determine the time to reach the end

We know that the maximum distance the puck can travel in the x-direction without bouncing is the length of the table (L). Therefore, we can use the following kinematic equation to find the time (t) it takes to travel this distance with the minimum initial velocity: L = v_min * t + 0.5 * a_x * t^2 (since the initial position is 0) We need to find the time (t) when the puck reaches W (width of the table) in the y-direction with the given acceleration (a). W = 0.5 * a_y * t^2 t = sqrt(2 * W/a)

Calculate the minimum initial velocity

Now, substitute the time (t) found in Step 3 back into the equation for x-direction motion in Step 2: L = v_min * sqrt(2 * W/a) + 0.5 * a_x * (2 * W/a) Solving for v_min, we get: v_min = (L - a_x * W)/sqrt(2 * W/a) Plug in the values: v_min = (2 - 2 * 1)/(sqrt(2 * 1/2)) v_min = 1/sqrt(2) = 0.707 m/s (rounded to three decimal places)

Draw the trajectories

Now that we have the minimum initial velocity, we can draw the trajectories for three different cases: v < v_min, v = v_min, and v > v_min. 1. When v < v_min, the puck will not reach the opposite short side without bouncing off one of the long sides. 2. When v = v_min (0.707 m/s), the puck will reach the opposite short side without bouncing. 3. When v > v_min, the puck will reach the opposite short side with room to spare before hitting the long sides. The trajectories can be drawn with arrows that follow these descriptions, showing the path of the puck in each case.

Key concepts

Minimum Initial Velocity

Understanding the minimum initial velocity is crucial in the realm of physics, especially when addressing problems in projectile motion. Minimum initial velocity refers to the least amount of speed that an object must have to reach a specific target without being influenced by other factors such as friction or air resistance. In the context of our air-hockey puck problem, it's the slowest speed at which the puck can be pushed to make it across the table while being accelerated by the rocket.

The problem presents a situation where the puck's trajectory must be calculated to ensure it doesn't collide with the table's long sides. To achieve this, we use kinematic equations to find the time it takes for the puck to reach the table's width and then apply this duration to decipher the needed velocity along the length. The formula yielded from the solution, \(v_{\min} = \frac{L - a_x \cdot W}{\sqrt{2 \cdot W/a}}\), encapsulates the motion in both x and y directions, giving us the precise initial velocity needed.

Projectile Motion

Projectile motion is a form of motion experienced by an object that is thrown near the Earth’s surface and moves along a curved path under the action of gravity only. However, in our air-hockey table scenario, an additional horizontal acceleration is imparted by the rocket, slightly altering the standard projectile motion.

In an ideal projectile motion situation, the only acceleration is due to gravity, acting downward. But with the rocket’s thrust here, we also have a horizontal acceleration to consider. Analyzing projectile motion involves breaking it down into two components: horizontal motion, which is uniform since no external horizontal forces are acting (ignoring the rocket thrust for a moment), and the vertical motion, which is uniformly accelerated due to gravity, or in this case, the rocket acceleration. This analysis helps predict where the puck will land under different initial velocities.

Acceleration

Acceleration is a vector quantity that represents the rate of change of velocity of an object. It is not just about how fast velocity changes, but also in what direction. In our exercise, the rocket's thrust provides a constant acceleration to the puck, pushing it along one axis of the table.

Since acceleration is constant, the kinematic equations simplify the solving process. We use one of these equations, \(W = 0.5 \cdot a_y \cdot t^2\), to find the time it takes for the puck to traverse the table width. It's important to note that when analyzing motion in only one dimension, acceleration can either speed up or slow down an object, but when dealing with two or more dimensions, it can also change the object's direction—crucial for trajectory analysis.

Trajectory Analysis

The trajectory of an object is the path that it follows as it moves through space. Trajectory analysis in physics allows us to predict this path by using the laws of classical mechanics. In the case of the air-hockey puck, the initial launch conditions, coupled with the force generated by the rocket, dictate its unique trajectory.

Through the given exercise, we learn that different initial velocities result in varying trajectories. The trajectory at minimum velocity is a sort of 'critical path' that allows the puck to barely make it across the table. If the initial velocity is less than this critical value, the puck does not make it across, and if it's greater, the puck will safely land with room to spare. By visualizing these different paths, as suggested in Step 5 of the solution, we can apply trajectory analysis to predict the successful movement of projectiles in different scenarios.