You're driving down the highway late one night at 18 m/s when a deer steps onto the road 42 m in front of you. Your reaction time before stepping on the brakes is 0.50 s , and the maximum deceleration of your car is 10 m/s2 .
A.)How much distance is between you and the deer when you come to a stop?
I got 17m.
B.)What is the maximum speed you could have and still not hit the deer?
I really don't know how to get Vmax.
Someone please help me. Thank You!
To calculate the maximum speed you could have and still not hit the deer, you need to consider the distance it takes for your car to come to a complete stop.
We can break down the problem into two parts:
1. Reaction time distance.
2. Braking distance.
1. Reaction time distance:
Given:
Initial velocity (v) = 18 m/s
Reaction time (t) = 0.50 s
The distance covered during the reaction time can be calculated using the formula:
d = v * t
Plugging in the values, we get:
d = 18 m/s * 0.50 s
d = 9 m
So, the distance covered during the reaction time is 9 meters.
2. Braking distance:
Given:
Maximum deceleration (a) = -10 m/s^2 (negative sign indicates deceleration, which is opposite to velocity)
Initial velocity (v) = 18 m/s
Final velocity (v') = 0 m/s
We can use the SUVAT equation to calculate the braking distance:
v'^2 = v^2 + 2ad
Rearranging the equation to solve for the braking distance (d):
d = (v'^2 - v^2) / (2a)
Plugging in the values, we get:
d = (0 m/s)^2 - (18 m/s)^2 / (2 * -10 m/s^2)
d = -324 m^2/s^2 / -20 m/s^2
d = 16.2 m
So, the braking distance is 16.2 meters.
Now, to find the maximum speed you could have and still not hit the deer, you add the reaction time distance and the braking distance:
Maximum speed = Reaction time distance + Braking distance
Maximum speed = 9 m + 16.2 m
Maximum speed = 25.2 m/s
Therefore, the maximum speed you could have and still not hit the deer is approximately 25.2 m/s.
To find the distance between you and the deer when you come to a stop, you can use the following steps:
1. Calculate the distance traveled during your reaction time:
- Since you're traveling at a constant speed, the distance traveled during the reaction time can be calculated as:
Distance = Speed * Time
Distance = 18 m/s * 0.50 s = 9 m
2. Calculate the distance required to come to a stop:
- The distance required to come to a stop can be calculated using the formula:
Distance = (Initial Velocity^2 - Final Velocity^2) / (2 * Acceleration)
- In this case, the initial velocity is 18 m/s, and the final velocity is 0 m/s (since you come to a stop).
Distance = (18^2 - 0^2) / (2 * -10 m/s^2) = 324 / -20 = -16.2 m
3. Calculate the total distance between you and the deer:
- The total distance between you and the deer is the sum of the distance traveled during reaction time and the distance required to come to a stop:
Total Distance = Distance Traveled + Distance Required = 9 m + (-16.2 m) = -7.2 m
Since distance cannot be negative, the result of -7.2 meters doesn't make sense in this context. Therefore, it's likely that there was an error in the calculations or the problem statement. Please double-check your numbers and the given information.
Now, let's move on to finding the maximum speed you could have and still not hit the deer (Vmax):
To find the maximum speed you could have and still not hit the deer, you need to consider the distance required to stop and your reaction time. The total distance required to stop is the sum of the distance traveled during reaction time and the distance required to come to a stop.
1. Calculate the distance traveled during your reaction time (same as step 1 above):
Distance = Speed * Time
Distance = 18 m/s * 0.50 s = 9 m
2. Calculate the distance required to come to a stop (same as step 2 above):
Distance = (Initial Velocity^2 - Final Velocity^2) / (2 * Acceleration)
Distance = (V^2 - 0^2) / (2 * -10 m/s^2) = V^2 / -20
3. Calculate the total distance required to stop:
Total Distance = Distance Traveled + Distance Required
Total Distance = 9 m + V^2 / -20
To not hit the deer, the total distance required to stop should be greater than or equal to the distance between you and the deer (42 m in this case). Therefore, the equation will be:
9 m + V^2 / -20 >= 42 m
To find Vmax, you need to solve for V in the above equation. Rearranging the equation, it becomes:
V^2 / -20 >= 42 m - 9 m
V^2 / -20 >= 33 m
To remove the negative sign, we can multiply both sides of the equation by -1:
V^2 / 20 <= -33 m
To isolate V^2, we can multiply both sides by 20:
V^2 <= -660 m
Since V^2 cannot be negative, the maximum speed you could have and still not hit the deer is not a real value. This indicates that you would need to slow down regardless of your speed in order to avoid hitting the deer.
Your speed from the time you hit the brakes is
v(t) = 18-10t
so, it takes you 1.8 seconds to stop
your distance from the time you hit the brakes is
s(t) = 18(t+0.5)-5t^2
so, s(1.8) = 18*2.3 - 5*1.8^2 = 25.2m
so, 17m looks like a good answer.
You want the maximum speed v such that s(t) < 42
you have speed = v-10t, so time to stop is v/10 seconds.
So, figure v such that s(t)=42 and anything less than that will stop before hitting the deer.
v(v/10+0.5) - 5(v/10)^2 = 42
v = 24.41 m/s