Answer :
Certainly! Here is a step-by-step solution to determine the positions and velocities of the two particles when their accelerations are equal.
### Step 1: Define the Position Functions
The positions of the two particles at any time [tex]\( t \)[/tex] are given by:
[tex]\[ s_1 = 4t - t^2 \][/tex]
[tex]\[ s_2 = 5t^2 - t^3 \][/tex]
### Step 2: Compute the Accelerations
Acceleration is the second derivative of position with respect to time. We need to find the second derivatives of [tex]\( s_1 \)[/tex] and [tex]\( s_2 \)[/tex].
For [tex]\( s_1 \)[/tex]:
[tex]\[ s_1 = 4t - t^2 \][/tex]
The first derivative (velocity) is:
[tex]\[ v_1 = \frac{ds_1}{dt} = 4 - 2t \][/tex]
The second derivative (acceleration) is:
[tex]\[ a_1 = \frac{d^2s_1}{dt^2} = -2 \][/tex]
For [tex]\( s_2 \)[/tex]:
[tex]\[ s_2 = 5t^2 - t^3 \][/tex]
The first derivative (velocity) is:
[tex]\[ v_2 = \frac{ds_2}{dt} = 10t - 3t^2 \][/tex]
The second derivative (acceleration) is:
[tex]\[ a_2 = \frac{d^2s_2}{dt^2} = 10 - 6t \][/tex]
### Step 3: Set the Accelerations Equal and Solve for [tex]\( t \)[/tex]
We need to find the time [tex]\( t \)[/tex] when the accelerations of both particles are equal:
[tex]\[ a_1 = a_2 \][/tex]
[tex]\[ -2 = 10 - 6t \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ 6t = 10 + 2 \][/tex]
[tex]\[ 6t = 12 \][/tex]
[tex]\[ t = 2 \][/tex]
### Step 4: Find the Positions of Both Particles at [tex]\( t = 2 \)[/tex]
Plug [tex]\( t = 2 \)[/tex] into the position functions to find the positions:
For [tex]\( s_1 \)[/tex]:
[tex]\[ s_1(2) = 4(2) - 2^2 \][/tex]
[tex]\[ s_1(2) = 8 - 4 \][/tex]
[tex]\[ s_1(2) = 4 \][/tex]
For [tex]\( s_2 \)[/tex]:
[tex]\[ s_2(2) = 5(2^2) - (2^3) \][/tex]
[tex]\[ s_2(2) = 5(4) - 8 \][/tex]
[tex]\[ s_2(2) = 20 - 8 \][/tex]
[tex]\[ s_2(2) = 12 \][/tex]
The positions of the particles when their accelerations are equal are [tex]\( s_1 = 4 \)[/tex] and [tex]\( s_2 = 12 \)[/tex].
### Step 5: Find the Velocities of Both Particles at [tex]\( t = 2 \)[/tex]
Plug [tex]\( t = 2 \)[/tex] into the velocity functions to find the velocities:
For [tex]\( v_1 \)[/tex]:
[tex]\[ v_1(2) = 4 - 2(2) \][/tex]
[tex]\[ v_1(2) = 4 - 4 \][/tex]
[tex]\[ v_1(2) = 0 \][/tex]
For [tex]\( v_2 \)[/tex]:
[tex]\[ v_2(2) = 10(2) - 3(2^2) \][/tex]
[tex]\[ v_2(2) = 20 - 12 \][/tex]
[tex]\[ v_2(2) = 8 \][/tex]
The velocities of the particles when their accelerations are equal are [tex]\( v_1 = 0 \)[/tex] and [tex]\( v_2 = 8 \)[/tex].
### Summary of Results
When the acceleration of the two particles is equal at [tex]\( t = 2 \)[/tex]:
- The positions of the particles are:
- [tex]\( s_1 = 4 \)[/tex]
- [tex]\( s_2 = 12 \)[/tex]
- The velocities of the particles are:
- [tex]\( v_1 = 0 \)[/tex]
- [tex]\( v_2 = 8 \)[/tex]
Thus, we have determined the positions and velocities of both particles at the time when their accelerations are equal.
### Step 1: Define the Position Functions
The positions of the two particles at any time [tex]\( t \)[/tex] are given by:
[tex]\[ s_1 = 4t - t^2 \][/tex]
[tex]\[ s_2 = 5t^2 - t^3 \][/tex]
### Step 2: Compute the Accelerations
Acceleration is the second derivative of position with respect to time. We need to find the second derivatives of [tex]\( s_1 \)[/tex] and [tex]\( s_2 \)[/tex].
For [tex]\( s_1 \)[/tex]:
[tex]\[ s_1 = 4t - t^2 \][/tex]
The first derivative (velocity) is:
[tex]\[ v_1 = \frac{ds_1}{dt} = 4 - 2t \][/tex]
The second derivative (acceleration) is:
[tex]\[ a_1 = \frac{d^2s_1}{dt^2} = -2 \][/tex]
For [tex]\( s_2 \)[/tex]:
[tex]\[ s_2 = 5t^2 - t^3 \][/tex]
The first derivative (velocity) is:
[tex]\[ v_2 = \frac{ds_2}{dt} = 10t - 3t^2 \][/tex]
The second derivative (acceleration) is:
[tex]\[ a_2 = \frac{d^2s_2}{dt^2} = 10 - 6t \][/tex]
### Step 3: Set the Accelerations Equal and Solve for [tex]\( t \)[/tex]
We need to find the time [tex]\( t \)[/tex] when the accelerations of both particles are equal:
[tex]\[ a_1 = a_2 \][/tex]
[tex]\[ -2 = 10 - 6t \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ 6t = 10 + 2 \][/tex]
[tex]\[ 6t = 12 \][/tex]
[tex]\[ t = 2 \][/tex]
### Step 4: Find the Positions of Both Particles at [tex]\( t = 2 \)[/tex]
Plug [tex]\( t = 2 \)[/tex] into the position functions to find the positions:
For [tex]\( s_1 \)[/tex]:
[tex]\[ s_1(2) = 4(2) - 2^2 \][/tex]
[tex]\[ s_1(2) = 8 - 4 \][/tex]
[tex]\[ s_1(2) = 4 \][/tex]
For [tex]\( s_2 \)[/tex]:
[tex]\[ s_2(2) = 5(2^2) - (2^3) \][/tex]
[tex]\[ s_2(2) = 5(4) - 8 \][/tex]
[tex]\[ s_2(2) = 20 - 8 \][/tex]
[tex]\[ s_2(2) = 12 \][/tex]
The positions of the particles when their accelerations are equal are [tex]\( s_1 = 4 \)[/tex] and [tex]\( s_2 = 12 \)[/tex].
### Step 5: Find the Velocities of Both Particles at [tex]\( t = 2 \)[/tex]
Plug [tex]\( t = 2 \)[/tex] into the velocity functions to find the velocities:
For [tex]\( v_1 \)[/tex]:
[tex]\[ v_1(2) = 4 - 2(2) \][/tex]
[tex]\[ v_1(2) = 4 - 4 \][/tex]
[tex]\[ v_1(2) = 0 \][/tex]
For [tex]\( v_2 \)[/tex]:
[tex]\[ v_2(2) = 10(2) - 3(2^2) \][/tex]
[tex]\[ v_2(2) = 20 - 12 \][/tex]
[tex]\[ v_2(2) = 8 \][/tex]
The velocities of the particles when their accelerations are equal are [tex]\( v_1 = 0 \)[/tex] and [tex]\( v_2 = 8 \)[/tex].
### Summary of Results
When the acceleration of the two particles is equal at [tex]\( t = 2 \)[/tex]:
- The positions of the particles are:
- [tex]\( s_1 = 4 \)[/tex]
- [tex]\( s_2 = 12 \)[/tex]
- The velocities of the particles are:
- [tex]\( v_1 = 0 \)[/tex]
- [tex]\( v_2 = 8 \)[/tex]
Thus, we have determined the positions and velocities of both particles at the time when their accelerations are equal.
