To solve this problem, we need to break it into two phases:

1. Phase 1: Reaction Time - During the reaction time of 0.50 seconds, the car continues moving at a constant speed before the brakes are applied.
2. Phase 2: Braking - After the reaction time, the car decelerates at a rate of $10 \, \text{m/s}^2$ until it comes to a stop.

Step 1: Distance covered during reaction time

The car is moving at 20 m/s before the brakes are applied, and during the reaction time of 0.50 seconds, it travels a distance given by: $d_{\text{reaction}} = v_{\text{initial}} \times t_{\text{reaction}}$ $d_{\text{reaction}} = 20 \, \text{m/s} \times 0.50 \, \text{s} = 10 \, \text{m}$

So, during the reaction time, the car travels 10 meters.

Step 2: Remaining distance for braking

The total distance to the deer is 60 meters. After the reaction time, the remaining distance for braking is: $d_{\text{braking}} = 60 \, \text{m} - 10 \, \text{m} = 50 \, \text{m}$

Step 3: Maximum speed at which the car can stop within the remaining distance

Using the kinematic equation for deceleration: $v_f^2 = v_i^2 + 2 a d$ where:

• $v_f = 0 \, \text{m/s}$ (final velocity when the car stops),
• $v_i$ is the initial velocity (the speed right before braking),
• $a = -10 \, \text{m/s}^2$ (deceleration),
• $d = 50 \, \text{m}$ (remaining distance).

Rearranging the equation to solve for $v_i$: $0 = v_i^2 + 2(-10)(50)$ $v_i^2 = 1000$ $v_i = \sqrt{1000} \approx 31.62 \, \text{m/s}$

Conclusion:

The maximum speed you could have and still not hit the deer is approximately 31.62 m/s.

Would you like more details on any part of the solution?

Here are 5 related questions:

1. What would happen if the deceleration was less than 10 m/s²?
2. How would the result change if your reaction time was longer?
3. What if the deer was closer, say 40 m away instead of 60 m?
4. How does the braking distance depend on speed in general?
5. What if the road conditions made the deceleration less effective, like in rain?

Tip: Always be aware of the total stopping distance, which includes both the reaction distance and the braking distance.