Find detailed and accurate answers to your questions on IDNLearn.com. Ask any question and receive accurate, in-depth responses from our dedicated team of experts.
Sagot :
To determine when object A overtakes object B, we need to find the time [tex]\( t \)[/tex] when their displacements are equal. Let's denote these displacements as [tex]\( s_A(t) \)[/tex] and [tex]\( s_B(t) \)[/tex] for objects A and B, respectively.
Firstly, we find the displacement functions for both objects by integrating their velocity functions.
### Velocity and Displacement of Object A
The velocity function of object A is given by:
[tex]\[ v_A(t) = \frac{t+8}{e} \][/tex]
To find the displacement [tex]\( s_A(t) \)[/tex], we integrate the velocity function:
[tex]\[ s_A(t) = \int_0^t v_A(u) \, du = \int_0^t \frac{u+8}{e} \, du \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \int_0^t (u + 8) \, du \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \left[ \frac{u^2}{2} + 8u \right]_0^t \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \left( \frac{t^2}{2} + 8t \right) \][/tex]
[tex]\[ s_A(t) = \frac{t^2}{2e} + \frac{8t}{e} \][/tex]
### Velocity and Displacement of Object B
The velocity function of object B is given by:
[tex]\[ v_B(t) = \frac{2t}{1+t^4} \][/tex]
To find the displacement [tex]\( s_B(t) \)[/tex], we integrate the velocity function, adding the 3-mile head start:
[tex]\[ s_B(t) = \int_0^t v_B(u) \, du + 3 = \int_0^t \frac{2u}{1+u^4} \, du + 3 \][/tex]
To solve this integral, let's use a substitution [tex]\( u^4 = x \)[/tex], hence [tex]\( du = \frac{1}{4} u^{-3/4} dx \)[/tex].
But, this can be complex and may necessitate numerical methods.
### Equating Displacements
Rather than solving the complex integral directly, we recognize that to find when [tex]\( s_A(t) = s_B(t) \)[/tex]:
[tex]\[ \frac{t^2}{2e} + \frac{8t}{e} = \int_0^t \frac{2u}{1+u^4} \, du + 3 \][/tex]
We can simplify using numerical methods or tools like Python's sympy for practical computation.
To solve the equation, we may simplify or use numerical computation:
### Calculating the Time
We use sympy to solve the equation:
```python
import sympy as sp
t = sp.symbols('t')
v_A = (t + 8) / sp.exp(1)
v_B = 2 t / (1 + t4)
s_A = sp.integrate(v_A, (t, 0, t))
s_B = sp.integrate(v_B, (t, 0, t)) + 3
# Find t when s_A equals s_B
solution = sp.solve(s_A - s_B, t)
# Ensure only positive solution
solution = [sol.evalf() for sol in solution if sol.evalf() > 0]
# Convert time to minutes
t_overtake_minutes = [float(sol 60) for sol in solution]
t_overtake_rounded = [round(minutes, 1) for minutes in t_overtake_minutes]
print(t_overtake_rounded)
```
The output will provide the time required in minutes.
Thus, after performing computations and rounding the time to one decimal place, we find the exact time when object A overtakes object B.
Firstly, we find the displacement functions for both objects by integrating their velocity functions.
### Velocity and Displacement of Object A
The velocity function of object A is given by:
[tex]\[ v_A(t) = \frac{t+8}{e} \][/tex]
To find the displacement [tex]\( s_A(t) \)[/tex], we integrate the velocity function:
[tex]\[ s_A(t) = \int_0^t v_A(u) \, du = \int_0^t \frac{u+8}{e} \, du \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \int_0^t (u + 8) \, du \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \left[ \frac{u^2}{2} + 8u \right]_0^t \][/tex]
[tex]\[ s_A(t) = \frac{1}{e} \left( \frac{t^2}{2} + 8t \right) \][/tex]
[tex]\[ s_A(t) = \frac{t^2}{2e} + \frac{8t}{e} \][/tex]
### Velocity and Displacement of Object B
The velocity function of object B is given by:
[tex]\[ v_B(t) = \frac{2t}{1+t^4} \][/tex]
To find the displacement [tex]\( s_B(t) \)[/tex], we integrate the velocity function, adding the 3-mile head start:
[tex]\[ s_B(t) = \int_0^t v_B(u) \, du + 3 = \int_0^t \frac{2u}{1+u^4} \, du + 3 \][/tex]
To solve this integral, let's use a substitution [tex]\( u^4 = x \)[/tex], hence [tex]\( du = \frac{1}{4} u^{-3/4} dx \)[/tex].
But, this can be complex and may necessitate numerical methods.
### Equating Displacements
Rather than solving the complex integral directly, we recognize that to find when [tex]\( s_A(t) = s_B(t) \)[/tex]:
[tex]\[ \frac{t^2}{2e} + \frac{8t}{e} = \int_0^t \frac{2u}{1+u^4} \, du + 3 \][/tex]
We can simplify using numerical methods or tools like Python's sympy for practical computation.
To solve the equation, we may simplify or use numerical computation:
### Calculating the Time
We use sympy to solve the equation:
```python
import sympy as sp
t = sp.symbols('t')
v_A = (t + 8) / sp.exp(1)
v_B = 2 t / (1 + t4)
s_A = sp.integrate(v_A, (t, 0, t))
s_B = sp.integrate(v_B, (t, 0, t)) + 3
# Find t when s_A equals s_B
solution = sp.solve(s_A - s_B, t)
# Ensure only positive solution
solution = [sol.evalf() for sol in solution if sol.evalf() > 0]
# Convert time to minutes
t_overtake_minutes = [float(sol 60) for sol in solution]
t_overtake_rounded = [round(minutes, 1) for minutes in t_overtake_minutes]
print(t_overtake_rounded)
```
The output will provide the time required in minutes.
Thus, after performing computations and rounding the time to one decimal place, we find the exact time when object A overtakes object B.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
