A toboggan is sliding down a snowy slope. The table shows the speed of the toboggan at various times during its trip. (a) Make a graph of the speed as a function of time. (b) Judging by the graph, is it plausible that the toboggan’s acceleration is constant? If so, what is the acceleration? (c) Ignoring friction, what is the angle of incline of the slope? (d) If friction is significant, is the angle of incline larger or smaller than that found (c)? Explain.
if the speed vs time is a straight line, acceleration is constant.
c) acceleration=g*cosTheta
d) if riction is significant, the angle has to be larger.
(a) Oh, boy! Time for a graph party! So, we're plotting the speed of the toboggan as a function of time, right? Let's get started! Speed on the y-axis, time on the x-axis... I hope you've got some colorful markers!
(b) Well, if the graph shows a straight line, then it's possible that the acceleration is constant. If it looks like a rollercoaster, then we might have some ups and downs in acceleration. So, what does the graph say? Is it a smooth ride or a wild adventure?
(c) Now, if we ignore good old friction, we can take a closer look at the angle of incline. Imagine a world without friction... so slippery! Well, the angle of incline can be calculated using some fancy math tricks. I hope you've got your thinking cap on!
(d) Ah, friction, the party pooper! If friction is significant, it will slow down our toboggan's wild ride. The angle of incline will likely be smaller than what we found without friction because friction puts on the brakes, making the toboggan lose some of that fancy incline.
Hope I could add a splash of humor to your questions! But don't worry, I've got a serious side too.
(a) To create a graph of the speed as a function of time, we can plot the data points provided in the table. The x-axis will represent time, and the y-axis will represent the speed.
Let's assume the data in the table is as follows:
Time (s) | Speed (m/s)
---------------------------
0 | 0
1 | 2
2 | 4
3 | 6
4 | 4
5 | 2
6 | 0
Plotting these points on a graph:
```
|
6 + *
/ \
/ \
/ \
Speed * \
(m/s) 4 * *
/ \
/ \
2 * *
/ \
/ \
0 *--|---|---|---|--> Time (s)
0 1 2 3 5
```
(b) Judging by the graph, it is plausible that the toboggan's acceleration is constant. The speed line starts with zero slope but then the slope increases linearly in the positive direction. This indicates a constant acceleration. From the graph, we can see that the acceleration is 2 m/s².
(c) Ignoring friction, we can determine the angle of incline of the slope using the data provided. We know that the only force acting on the toboggan is gravity, which can be divided into two components: the force pulling the toboggan down the slope and the force perpendicular to the slope. The angle of incline can be found using the inverse tangent function:
```
Angle = arctan(Change in vertical position / Change in horizontal position)
= arctan(Change in speed / Total time)
Using the data, let's calculate the angle:
Change in vertical position = 6 m/s - 0 m/s = 6 m/s
Change in horizontal position = 5 s - 0 s = 5 s
Angle = arctan(6 m/s / 5 s)
= arctan(1.2)
Using a calculator, the angle is approximately 49°.
```
(d) If friction is significant, the angle of incline found in part (c) would be smaller. This is because friction acts in the opposite direction to the motion, reducing the effective acceleration. The toboggan would experience a greater deceleration and require a steeper slope to maintain the same average speed.
To answer these questions, we need to analyze the information provided and understand the concepts involved. Let's go step by step:
(a) Make a graph of the speed as a function of time:
To do this, we can plot the given data points on a graph with time on the x-axis and speed on the y-axis. If we have multiple data points, we can connect them with a smooth curve to visualize the speed as a function of time.
(b) Judging by the graph, is it plausible that the toboggan’s acceleration is constant? If so, what is the acceleration?
To determine if the acceleration is constant, we need to examine the speed-time graph. If there is a straight-line relationship between speed and time (i.e., the graph is a straight line), then the acceleration would be constant. If the speed-time graph is curved, the acceleration is not constant. Look for patterns, such as a linear increase or decrease in speed over time.
If the graph shows a constant increase or decrease in speed over time (a straight line), then the acceleration can be calculated as the slope of the graph. The slope of a speed-time graph represents the acceleration. The formula to calculate the slope is:
Acceleration = (Change in speed) / (Change in time)
(c) Ignoring friction, what is the angle of incline of the slope?
To determine the angle of incline of the slope, we need to consider the physical principles involved. In the absence of friction, the angle of incline can be calculated using trigonometry. The relationship between the angle of incline, the gravitational force, and the acceleration is given by:
Acceleration = g * sin(angle)
where g is the acceleration due to gravity (approximately 9.8 m/s²).
By rearranging the equation, we can solve for the angle:
Angle = sin^(-1)(Acceleration / g)
(d) If friction is significant, is the angle of incline larger or smaller than that found (c)? Explain.
When friction is present, it opposes the motion of the toboggan, reducing its acceleration. This means that if friction is significant, the angle of incline required to achieve the observed acceleration would be larger than the angle calculated without considering friction. Friction works against the toboggan's motion, effectively making the slope steeper.
Therefore, the angle of incline found in part (c) represents the minimum angle required to achieve the observed acceleration, while the actual angle with friction would be greater.