7. A cyclist is travelling with a speed of 12.0 m/s when she applies the brakes. After slowing for 3.0 s, her speed has been reduced to 4.0 m/s.
(a) What distance does she travel during this time?
(b) What is her acceleration?
(c) If she continues braking, how much longer will it take her to stop?
(d) If she continues to travel with her new speed, how far will she travel during the next 3.0 s?
avg speed= 8m/s distance=avgspeed*3sec
b. a=change speed/time
c. time=4.0/a
d. distnace=4m/s * 3 sec
(a) To find the distance traveled, we can use the formula:
\[d = \frac{1}{2}(v_f + v_i)(t)\]
Where:
d is the distance traveled
v_f is the final velocity
v_i is the initial velocity
t is the time taken
Given:
v_i = 12.0 m/s
v_f = 4.0 m/s
t = 3.0 s
Plugging the values into the formula:
\[d = \frac{1}{2}(4.0 + 12.0)(3.0)\]
\[d = \frac{16.0}{2}(3.0)\]
\[d = 8.0(3.0)\]
\[d = 24.0\text{ meters}\]
Therefore, she travels a distance of 24.0 meters during this time.
(b) To find the acceleration, we can use the formula:
\[a = \frac{v_f - v_i}{t}\]
Given:
v_i = 12.0 m/s
v_f = 4.0 m/s
t = 3.0 s
Plugging the values into the formula:
\[a = \frac{4.0 - 12.0}{3.0}\]
\[a = \frac{-8.0}{3.0}\]
\[a = -2.67 \approx \text{-}2.7 \, \text{m/s}^2\]
Therefore, her acceleration is approximately -2.7 m/s^2.
(c) To determine how much longer it will take her to stop, we need to find the time it takes for her to reach a speed of 0 m/s. This can be calculated using the formula:
\[t = \frac{v_f - v_i}{a}\]
Given:
v_i = 12.0 m/s
v_f = 0.0 m/s
a = -2.7 m/s^2
Plugging the values into the formula:
\[t = \frac{0.0 - 12.0}{-2.7}\]
\[t = \frac{-12.0}{-2.7}\]
\[t = 4.44 \, \text{s}\]
Therefore, it will take her approximately 4.44 seconds to stop completely.
(d) To find the distance traveled during the next 3.0 seconds while maintaining a speed of 4.0 m/s, we can use the formula:
\[d = (v_i)(t)\]
Given:
v_i = 4.0 m/s
t = 3.0 s
Plugging the values into the formula:
\[d = (4.0)(3.0)\]
\[d = 12.0 \, \text{meters}\]
Therefore, she will travel a distance of 12.0 meters during the next 3.0 seconds.
(a) To find the distance traveled during the given time, we can use the formula:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is 12.0 m/s, the time is 3.0 s, and the final velocity is 4.0 m/s (since the cyclist slows down). We don't have the acceleration yet, so let's solve for it first.
(b) To find the acceleration, we can use the formula:
acceleration = (final velocity - initial velocity) / time
Plugging in the values, we have:
acceleration = (4.0 m/s - 12.0 m/s) / 3.0 s
(c) To find out how much longer it will take her to stop, we need to calculate the time it takes for her to come to a complete stop. When the cyclist comes to a complete stop, her final velocity will be 0 m/s. We know the acceleration from part (b), so we can use the formula:
final velocity = initial velocity + acceleration * time
Since the final velocity is 0 m/s and the initial velocity is 4.0 m/s, we have:
0 m/s = 4.0 m/s + acceleration * time
Solving for time, we can rearrange the equation and substitute the known values:
time = -4.0 m/s / acceleration
(d) Finally, to find the distance she will travel during the next 3.0 s with her new speed of 4.0 m/s, we can use the formula:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the initial velocity is 4.0 m/s, the time is 3.0 s, and the acceleration is 0 m/s^2 since she has already come to a stop.