A stone is thrown horizontally at 30.0 m/s from the top of a very tall cliff. (a) Calculate its horizontal position and vertical position at 2-s intervals for the first 10.0s. (b) Plot your positions from part (a) to scale. Then connect your points with a smooth curve to show the trajectory of the stone.
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To solve this problem, we can use the equations of motion for both horizontal and vertical directions. Let's break it down into two parts:
(a) Calculating the horizontal and vertical positions at 2-second intervals:
First, let's consider the horizontal motion. Since the stone is thrown horizontally, its initial horizontal velocity will remain constant throughout its motion. The horizontal position can be calculated using the formula:
Horizontal distance = Initial horizontal velocity * Time
In this case, the initial horizontal velocity is 30.0 m/s, and we are interested in the position at 2-second intervals for the first 10 seconds. So we can calculate the horizontal position at each interval:
At t = 2 s: Horizontal distance = (30.0 m/s) * (2 s) = 60 m
At t = 4 s: Horizontal distance = (30.0 m/s) * (4 s) = 120 m
At t = 6 s: Horizontal distance = (30.0 m/s) * (6 s) = 180 m
...
So on until t = 10 s.
Now let's consider the vertical motion. The vertical position of the stone can be calculated using the equations of motion under constant acceleration. The acceleration in this case is due to gravity, which is approximately 9.8 m/s^2. The formulas for vertical motion are:
Vertical distance = Initial vertical velocity * Time + (1/2) * Acceleration * Time^2
Initial vertical velocity = 0 m/s since the stone is thrown horizontally, not vertically.
Using these formulas, we can calculate the vertical position at each 2-second interval:
At t = 2s: Vertical distance = 0 * (2 s) + (0.5) * (9.8 m/s^2) * (2 s)^2 = 19.6 m
At t = 4s: Vertical distance = 0 * (4 s) + (0.5) * (9.8 m/s^2) * (4 s)^2 = 78.4 m
At t = 6s: Vertical distance = 0 * (6 s) + (0.5) * (9.8 m/s^2) * (6 s)^2 = 176.4 m
...
So on until t = 10 s.
(b) Plotting the positions and connecting them with a smooth curve:
To plot the positions, create a graph with two axes - one for the horizontal position and one for the vertical position. Mark the positions calculated at each interval on the respective axes. Then connect the points smoothly using a curve to show the trajectory of the stone.
Make sure to label each axis with the appropriate units (meters in this case) and title the graph to indicate it represents the trajectory of the stone thrown horizontally.
By following this process, you should be able to calculate and plot the horizontal and vertical positions of the stone at 2-second intervals for the first 10 seconds and display its trajectory on a graph.