Question

A marble is rolled so that it is projected horizontally off the top landing of a staircase. The initial speed of the marble is $3.0 \mathrm{m} / \mathrm{s} .\( Each step is \)0.18 \mathrm{m}\( high and \)0.30 \mathrm{m}$ wide. Which step does the marble strike first?

Short answer

Answer: To find out which step the marble will strike first, we can follow the step-by-step solution mentioned above to break down the marble's motion into vertical and horizontal components, and then use the provided equations to calculate the marble's behavior for each step. By iterating through each step and comparing the horizontal distance traveled by the marble with the width of the step, we can determine the first step the marble strikes.

Step-by-step solution

Determine the equation for the vertical motion of the marble

Determine the time t it takes for the marble to fall down a vertical height of h by using the equation \(h = \frac{1}{2}gt^2\), where g is acceleration due to gravity (9.8 \( m/s^2 \)).

Determine the equation for the horizontal motion of the marble

Determine the horizontal distance x the marble travels in time t by using the equation \(x = v_0t\), where \(v_0\) is the initial horizontal velocity of the marble.

Iterate through each step until the marble strikes a step

For each step, starting from the first step, calculate the time t it takes for the marble to reach the height of that step using the vertical motion equation. Then, calculate the horizontal distance the marble travels in that time using the horizontal motion equation. Compare this horizontal distance with the width of the step to check if the marble strikes the step: 1. Calculate the vertical height of the nth step: \(h_n = n \times 0.18m\) 2. Calculate the time it takes for the marble to reach the height of the nth step: \(t_n = \sqrt{\frac{2h_n}{g}}\) 3. Calculate the horizontal distance the marble travels in time \(t_n\): \(x_n = v_0t_n\) 4. Check if the marble strikes the nth step: \(x_n < n \times 0.30m\) If the marble strikes the nth step, the process is stopped. Otherwise, move on to the next step n+1 and repeat the process.

Report the step the marble strikes first

Once the step where the marble strikes first is identified using the iterative process in Step 3, report the result.