We need to find the time it takes for the motorcycle to catch up with the car after the motorcycle starts accelerating.

Given:

• Initial speed of both the car and the motorcycle: $v_0 = 19.5 \, \text{m/s}$
• Initial distance between the car and the motorcycle: $d_0 = 90.0 \, \text{m}$
• The motorcycle starts accelerating after $t_1 = 3.00 \, \text{s}$.
• Acceleration of the motorcycle: $a = 4.00 \, \text{m/s}^2$

To find:

• The time $\Delta t = t_2 - t_1$, where $t_2$ is the time when the motorcycle catches up with the car after the motorcycle starts accelerating.

Step-by-step solution:

1. Position of the car: The car is moving at a constant speed, so its position at any time $t$ after $t_1$ can be described by: $x_{\text{car}}(t) = v_0 \cdot t$

2. Position of the motorcycle: After the motorcycle starts accelerating at $t = t_1$, its position at any time $t$ after it starts accelerating is: $x_{\text{motorcycle}}(t) = v_0 \cdot t + \frac{1}{2} a \cdot (t - t_1)^2$ where $a = 4.00 \, \text{m/s}^2$ is the acceleration.

3. Catching up condition: The motorcycle catches up to the car when their positions are the same: $x_{\text{motorcycle}}(t_2) = x_{\text{car}}(t_2)$ Substituting the expressions for $x_{\text{car}}(t)$ and $x_{\text{motorcycle}}(t)$ into this equation gives: $v_0 \cdot t_2 + \frac{1}{2} a \cdot (t_2 - t_1)^2 = v_0 \cdot t_2$

4. Solve for $t_2 - t_1$: Since $v_0 \cdot t_2$ cancels out on both sides, we're left with: $\frac{1}{2} a \cdot (t_2 - t_1)^2 = d_0$ Substituting the given values $a = 4.00 \, \text{m/s}^2$ and $d_0 = 90.0 \, \text{m}$, we have: $\frac{1}{2} \cdot 4.00 \cdot (t_2 - t_1)^2 = 90.0$ Simplifying: $2.00 \cdot (t_2 - t_1)^2 = 90.0$ $(t_2 - t_1)^2 = \frac{90.0}{2.00}$ $(t_2 - t_1)^2 = 45.0$ Taking the square root: $t_2 - t_1 = \sqrt{45.0} \approx 6.71 \, \text{seconds}$

Final answer:

The time it takes for the motorcycle to catch up with the car after it starts accelerating is approximately $6.71 \, \text{seconds}$.

Would you like more details or have any questions?

Here are some related questions you could explore:

1. What is the distance traveled by the motorcycle while catching up?
2. How fast is the motorcycle traveling when it catches up?
3. What if the motorcycle accelerated at a different rate?
4. How much total time does it take for the motorcycle to catch up, including the first 3 seconds?
5. What if the initial distance was different?

Tip: When analyzing motion problems, breaking them into phases (constant speed, acceleration, etc.) makes solving much easier!