An ant is crawling along the x-axis such that the graph of its position on the x-axis
versus time is a semi-circle (see figure). The total distance covered in the 4 s is
x[m]
4 t [sI
no figure, no question.
To determine the total distance covered by the ant in 4 seconds, we first need to find the equation of the graph representing its position on the x-axis.
Since the graph is a semi-circle, it means the ant is moving in one direction until it reaches the maximum point, and then it starts moving in the opposite direction.
Let's assume the radius of the semi-circle is r. The equation of a semi-circle centered at the origin (0,0) can be written as:
x^2 + y^2 = r^2
Since the ant is only moving along the x-axis, the y-coordinate is always 0. Therefore, the equation becomes:
x^2 + 0^2 = r^2
x^2 = r^2
Simplifying, we get:
x = ± r
The ant starts at the leftmost point of the semi-circle, which is (-r, 0). As it crawls along the x-axis, it reaches the rightmost point, which is (r, 0). At this point, it turns around and starts moving towards the left.
The total distance covered by the ant can be calculated by finding the distance between the starting point and the turning point, and then doubling it.
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the starting point is (-r, 0) and the turning point is (r, 0). Plugging in the values, we get:
d = √((r - (-r))^2 + (0 - 0)^2)
d = √((2r)^2)
d = √(4r^2)
d = 2r
The total distance covered by the ant in the 4 seconds is twice the radius of the semi-circle:
Total distance = 2 * r
Please provide the value of the radius (r) to compute the total distance covered by the ant.
To find the total distance covered by the ant in 4 seconds, we need to calculate the length of the semi-circular path.
In a semi-circle, the distance traveled is equal to half the circumference of the full circle.
The formula for the circumference of a circle is given by:
Circumference = 2 * π * radius
Since the graph represents a semi-circle, we need to find the radius of the full circle. The radius is the distance between the ant's starting position and the highest point (vertex) of the semi-circle.
Let's assume that the highest point of the semi-circle is at coordinates (0, R). The ant starts at coordinates (0, 0).
From the given information, we know that the highest point is reached at 2 seconds, which means the ant travels for 2 seconds to reach that point.
Using the time-distance graph, we can determine that the velocity of the ant is constant and equal to:
velocity = (R - 0) / (2 - 0)
= R / 2
Since velocity is equal to distance divided by time, we can rewrite the equation as:
distance = velocity * time
distance = (R / 2) * 2
distance = R
So, the radius of the full circle is equal to R, which is the distance covered by the ant in 2 seconds.
Now, we can calculate the circumference of the full circle using the formula mentioned earlier:
Circumference = 2 * π * R
Since we only need half of the circumference for the semi-circle, the distance covered by the ant in the 4 seconds is:
Total distance = (1/2) * Circumference
= (1/2) * (2 * π * R)
= π * R
Therefore, the total distance covered by the ant in 4 seconds is π * R.