Question

A diver runs horizontally off the end of a \(3.0-\mathrm{m}\)-high diving board with an initial speed of \(1.8 \mathrm{~m} / \mathrm{s}\). (a) Given that the diver's initial position is \(x_{\mathrm{i}}=0\) and \(y_{\mathrm{i}}=3.0 \mathrm{~m}\), find her \(x\) and \(y\) positions at the times \(t=0.25 \mathrm{~s}\), \(t=0.50 \mathrm{~s}\), and \(t=0.75 \mathrm{~s}\). (b) Plot the results from part (a), and sketch the corresponding parabolic path.

Short answer

Positions: (0.45m, 2.692m), (0.90m, 2.225m), (1.35m, 1.598m). Plot trajectory to see a parabolic path.

Step-by-step solution

Calculate Horizontal Position

The horizontal position \( x \) of the diver can be found using the formula: \( x = x_i + v_{x} \, t \). Here, \( x_i = 0 \) and \( v_{x} = 1.8 \, \mathrm{m/s} \). - At \( t = 0.25 \mathrm{~s} \), \( x = 0 + 1.8 \times 0.25 = 0.45 \mathrm{~m} \).- At \( t = 0.50 \mathrm{~s} \), \( x = 0 + 1.8 \times 0.50 = 0.90 \mathrm{~m} \).- At \( t = 0.75 \mathrm{~s} \), \( x = 0 + 1.8 \times 0.75 = 1.35 \mathrm{~m} \).

Calculate Vertical Position

The vertical position \( y \) is determined by the equation: \( y = y_i - \frac{1}{2} g t^2 \), where \( y_i = 3.0 \mathrm{~m} \) and \( g = 9.8 \mathrm{~m/s^2} \). - At \( t = 0.25 \mathrm{~s} \), \( y = 3.0 - \frac{1}{2} \times 9.8 \times (0.25)^2 = 2.692 \mathrm{~m} \).- At \( t = 0.50 \mathrm{~s} \), \( y = 3.0 - \frac{1}{2} \times 9.8 \times (0.50)^2 = 2.225 \mathrm{~m} \).- At \( t = 0.75 \mathrm{~s} \), \( y = 3.0 - \frac{1}{2} \times 9.8 \times (0.75)^2 = 1.598 \mathrm{~m} \).

Plot the Results

Using the calculated \( x \) and \( y \) positions from Steps 1 and 2, plot the points on a graph. Connect these points to reflect the trajectory of the dive. The path should appear as a parabola, illustrating the diver's projectile motion as parabolic.

Key concepts

Horizontal Motion

When discussing projectile motion, horizontal motion is one of the simplest components to understand. Imagine a diver running straight off a diving board. The motion directly forward is what we call horizontal motion. This occurs on a flat plane, and in the absence of air resistance, it's consistent.

In our example, the diver moves off the board at a speed of \(1.8\,\mathrm{m/s}\). If there are no other horizontal forces acting on her, she will continue to move at that speed. We calculate this movement using the formula \(x = x_i + v_x\cdot t\).

For each time increment (e.g., \(t = 0.25\) seconds, \(t = 0.50\) seconds), we multiply the speed \(v_x\) by time \(t\) to get her horizontal position.

Vertical Motion

Vertical motion in projectile trajectory involves gravity, which pulls everything toward the earth at \(9.8\,\mathrm{m/s^2}\). Unlike horizontal motion, vertical motion is subject to changing speeds as gravity accelerates the moving object.

Begin by using the formula: \(y = y_i - \frac{1}{2}gt^2\). Here, the negative sign indicates the downward pull of gravity. The diver starts at a height, so her motion involves decreasing her vertical position over time. As seconds pass (0.25, 0.50, 0.75), you can see her height decrease, which is why the distance her body travels vertically reduces as time goes on.

This creates a parabolic shape in her path, showcasing how both horizontal and vertical components combine to shape the trajectory.

Kinematics

Kinematics is the study of motion without considering the forces causing it. In this specific scenario, understanding kinematics helps in predicting where the diver will be at certain points in time.

Using equations from kinematics, we calculate both horizontal and vertical motions independently. Important formulas include: for horizontal motion \(x = x_i + v_xt\) and for vertical motion \(y = y_i - \frac{1}{2}gt^2\).

These calculations show how the diver's path is determined by her initial velocity, gravity, and time. All these kinematic equations are crucial in visualizing and plotting the path of any projectile, such as our diver, on a graph.

Physics Problem Solving

Solving physics problems involves a methodical approach. Start by identifying what you know and what you need to find out. With our diver, we know her initial speed, position, and the force of gravity.

Next, write down the relevant equations. In this case, you used horizontal motion formulae to find her movement across the board and vertical motion formulae to calculate how far she falls over time. Incorporate these into your calculations step by step.

The important part of solving these problems is to maintain clarity and consistency. Check your units, ensure your math is accurate, and don't rush. Each answer builds on the previous one, revealing how physics unravels nature's mysteries. With practice, such problems become much easier and more rewarding to solve.