## Related questions with answers

When freeways were first built in the United States in the $1950$s, entrance ramps were often too short for an entering car to safely merge into existing traffic. Consider an aggressive car acceleration of $a=4.0 \mathrm{~m} / \mathrm{s}^2$ and an initial car speed of $v_0=25 \mathrm{mi} / \mathrm{h}$ as the car enters the entrance ramp of a freeway where other cars are moving at $55 \mathrm{mi} / \mathrm{h}$. $(a)$ If the ramp has a length of $d=40 \mathrm{yd}$, what is the car's speed $v$ in miles per hour as it attempts to merge? $(b)$ What is the minimum length $d$ in yards needed for the car's speed to match the speed of the other cars?

Solution

Verified**Givens**

- $a=4\ \frac{\text{m}}{\text{s}^2}$ - the acceleration of the car
- $v_0=25\ \frac{\text{mi}}{\text{h}}=11.2\ \frac{\text{m}}{\text{s}}$ - the initial speed of the car
- $v_c=55\ \frac{\text{mi}}{\text{h}}=24.6\ \frac{\text{m}}{\text{s}}$ - the speed of the other cars
- $d=40\ \text{yd}=36\ \text{m}$ - the length of the ramp

**Required**

- a) $v(t=\tau)$ - the speed of the car when it reaches the end of the ramp
- b) $d_\text{min}$ - the minimum length of the ramp

*How can we use the known expressions for the speed and the distance travelled by the car in order to solve this problem?*

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