Carlos and Katherine are estimating acceleration by rolling a ball from rest down a ramp. At 1 second, the ball is moving at 5 meters per second (m/s), at 2 seconds, the ball is moving at 10 m/s, at 3 seconds, the ball is moving at 15 m/s, and at 4 seconds, it is moving at 20 m/s. When graphed on an xy-plane, which equation best describes the ball's estimated acceleration where y expresses speed and x expresses time?

Options

A. `y=5x+5`
B. `y=25x`
C. `y=5x`
D. `y=(4x+1)^(2)+5`

Answer

To solve the problem step by step, we will analyze the motion of the ball and determine the relationship between speed and time.

### Step 1: Understand the given data
The problem provides the speed of the ball at different time intervals:
- At 1 second: 5 m/s
- At 2 seconds: 10 m/s
- At 3 seconds: 15 m/s
- At 4 seconds: 20 m/s

### Step 2: Identify the relationship
We need to find the relationship between speed (y) and time (x). The speed increases uniformly as time increases, suggesting a linear relationship.

### Step 3: Plot the points
We can plot the points on an xy-plane:
- (1, 5)
- (2, 10)
- (3, 15)
- (4, 20)

### Step 4: Determine the type of graph
Since the points form a straight line and pass through the origin (0, 0), we can conclude that the relationship is linear.

### Step 5: Find the slope of the line
To find the equation of the line, we need to calculate the slope (m). We can use two points, for example, (1, 5) and (2, 10).

Using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \]

### Step 6: Write the equation of the line
Since the line passes through the origin, the equation can be written in the slope-intercept form:
\[ y = mx \]
Substituting the slope we found:
\[ y = 5x \]

### Step 7: Conclusion
The equation that best describes the ball's estimated acceleration, where y expresses speed and x expresses time, is:
\[ y = 5x \]