Gabriele enters an east-w straight bike path at the 3.0-km mark and rides w at a constant speed of 8.0 m/s. At the same time, Xena rides east from the1.0-km mark at a constant speed of 6.0 m/s.
1. Write function x(t) that describe the position of Gabriele as a function of time with respect to Earth.
2. Write function x(t) that describe the position of Xena as a function of time with respect to Earth.
3. Where do they meet each other?
4. Write a function x(t) that describes Xena's motion with respect to Gabriele.
1. The function x(t) that describes the position of Gabriele as a function of time with respect to Earth is:
x(t) = 3.0 km + (8.0 m/s) * t
2. The function x(t) that describes the position of Xena as a function of time with respect to Earth is:
x(t) = 1.0 km + (6.0 m/s) * t
3. To find where they meet each other, we need to set their positions equal to each other and solve for time:
3.0 km + (8.0 m/s) * t = 1.0 km + (6.0 m/s) * t
Simplifying this equation, we get:
2.0 km = (2.0 m/s) * t
t = 1.0 km / (2.0 m/s)
t = 0.5 hours
Therefore, they meet each other after 0.5 hours of travel.
4. The function x(t) that describes Xena's motion with respect to Gabriele is:
x(t) = (1.0 km + (6.0 m/s) * t) - (3.0 km + (8.0 m/s) * t)
x(t) = -2.0 km - (2.0 m/s) * t
1. To write the function x(t) that describes the position of Gabriele as a function of time with respect to Earth, we can use the formula for position (x) as a function of time (t) for an object moving at a constant speed. In this case, Gabriele is moving east at a constant speed of 8.0 m/s, starting from the 3.0-km mark. Therefore, the function would be:
x(t) = 3.0 km + (8.0 m/s) * t
Here, t represents the time elapsed in seconds since Gabriele started moving.
2. Similarly, for Xena's position as a function of time with respect to Earth, she is also moving east, but at a constant speed of 6.0 m/s, starting from the 1.0-km mark. Hence, the function would be:
x(t) = 1.0 km + (6.0 m/s) * t
3. To find where Gabriele and Xena meet each other, we need to equate their positions at a certain time. Let's assume they meet after time t. So, at that time:
Gabriele's position = Xena's position
3.0 km + (8.0 m/s) * t = 1.0 km + (6.0 m/s) * t
Simplifying the equation, we get:
2.0 km = (2.0 m/s) * t
Solving for t, we find:
t = 1.0 s
Therefore, they meet each other after 1.0 second.
4. To write the function x(t) that describes Xena's motion with respect to Gabriele, we subtract Gabriele's position as a function of time from Xena's position as a function of time. Therefore, the function would be:
x(t) = (1.0 km + (6.0 m/s) * t) - (3.0 km + (8.0 m/s) * t)
Simplifying the equation, we get:
x(t) = (1.0 km - 3.0 km) + (6.0 m/s - 8.0 m/s) * t
x(t) = -2.0 km + (-2.0 m/s) * t
Here, the negative sign indicates that Xena is behind Gabriele when their positions are compared.
1. To find the function x(t) that describes the position of Gabriele as a function of time with respect to Earth, we can use the equation:
x(t) = x₀ + v*t
Here, x₀ represents the initial position of Gabriele on the bike path, which is 3.0 km. v represents the velocity of Gabriele, which is 8.0 m/s. And t represents the time elapsed since Gabriele started riding.
So the function x(t) for Gabriele would be:
x(t) = 3.0 km + (8.0 m/s) * t
2. Similarly, to find the function x(t) that describes the position of Xena as a function of time with respect to Earth, we can use the same equation:
x(t) = x₀ + v*t
In this case, x₀ represents the initial position of Xena on the bike path, which is 1.0 km. v represents the velocity of Xena, which is 6.0 m/s. And t represents the time elapsed since Xena started riding.
So the function x(t) for Xena would be:
x(t) = 1.0 km + (6.0 m/s) * t
3. To find the point where they meet each other, we need to equate the functions for Gabriele and Xena:
3.0 km + (8.0 m/s) * t = 1.0 km + (6.0 m/s) * t
Now, let's solve for t:
(8.0 m/s - 6.0 m/s) * t = 1.0 km - 3.0 km
2.0 m/s * t = -2.0 km
Since the distance cannot be negative, there is no real solution to this equation, which means Gabriele and Xena do not meet each other.
4. Finally, let's find the function x(t) that describes Xena's motion with respect to Gabriele. This can be determined by subtracting Gabriele's position from Xena's position at any given time:
x(t) = (1.0 km + (6.0 m/s) * t) - (3.0 km + (8.0 m/s) * t)
Simplifying this equation, we get:
x(t) = -2.0 km - (2.0 m/s) * t