A bike takes 9.0 seconds to decelerate to a stop over a distance of 30 meters. How fast was the bike traveling before the biker started to decelerate?
To find the initial speed of the bike before deceleration, we can use the equation:
\[v_f^2 = v_i^2 + 2ad\]
where:
\(v_f\) = final velocity (which is 0 m/s since the bike comes to a stop)
\(v_i\) = initial velocity (what we want to find)
\(a\) = acceleration (since the bike is decelerating, the acceleration is negative)
\(d\) = distance traveled (30 meters)
Plugging in the given values, we get:
\(0^2 = v_i^2 + 2(-a)(30)\)
Simplifying, we have:
\(0 = v_i^2 - 60a\)
Since we know that the time it takes to decelerate to a stop is 9.0 seconds, we can use the equation:
\(a = \frac{{\Delta v}}{{\Delta t}}\)
to find the acceleration. In this case, the change in velocity is \(0 - v_i = -v_i\) and the change in time is 9.0 seconds. Substituting these values into the equation, we get:
\(a = \frac{{-v_i}}{{9.0}}\)
Plugging this value of acceleration back into the initial equation, we have:
\(0 = v_i^2 - 60 \left(\frac{{-v_i}}{{9.0}}\right)\)
Simplifying further:
\(0 = v_i^2 + \frac{{-60v_i}}{{9.0}}\)
Multiplying through by 9.0 to remove the fraction:
\(0 = 9.0v_i^2 - 60v_i\)
Rearranging the equation:
\(9.0v_i^2 - 60v_i = 0\)
Factoring out \(v_i\):
\(v_i(9.0v_i - 60) = 0\)
Setting each factor equal to zero and solving for \(v_i\):
\(v_i = 0\) (which is not possible since the bike was definitely moving)
\(v_i = \frac{{60}}{{9.0}}\)
Using a calculator, we can find that \(v_i \approx 6.67\) m/s.
Therefore, the bike was traveling at approximately 6.67 meters per second (m/s) before the biker started to decelerate.
To find the initial speed of the bike before deceleration, we can use the formula of uniformly decelerating motion:
v^2 = u^2 - 2aS
where:
v = final speed (0 m/s, since the bike comes to a stop)
u = initial speed (what we need to find)
a = acceleration (which can be calculated from the given information)
S = distance traveled (30 meters)
Since the bike comes to a stop, the final speed (v) is 0 m/s. Rearranging the formula, we have:
u^2 = 2aS
We are given that the distance traveled (S) is 30 meters. Now we need to find the acceleration (a). The deceleration is defined as the negative acceleration, so:
a = -v / t
where:
v = final speed (0 m/s)
t = time taken (9.0 seconds)
Substituting the given values, we get:
a = -0 / 9.0
Since the bike comes to a stop, the acceleration (a) is 0 m/s^2. Substituting this into the formula for initial speed (u), we have:
u^2 = 2(0)(30)
Simplifying further, we have:
u^2 = 0
To find the initial speed (u), we take the square root of both sides:
u = sqrt(0)
u = 0 m/s
Therefore, the bike was traveling at 0 m/s before the biker started to decelerate.
To find the initial speed of the bike before deceleration, we can use the equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the bike comes to a stop)
u = initial velocity (what we want to find)
a = acceleration (negative value because it's deceleration)
s = distance traveled (30 meters)
Since the final velocity is 0 and the distance traveled is 30 meters, we can rearrange the equation to solve for the initial velocity:
u^2 = v^2 - 2as
Substituting the known values:
u^2 = 0^2 - 2(-a)(s)
u^2 = 2as
Now, we can substitute the given values. The distance traveled is 30 meters, and it takes 9.0 seconds to come to a stop. So, the deceleration, a, can be calculated as:
a = (change in velocity) / (time)
a = (0 - u) / 9.0
Since the bike is decelerating to a stop, the final velocity is 0.
Substituting and solving:
u^2 = 2 * [(0 - u) / 9.0] * 30
Simplify:
u^2 = (2 * 30 * u) / 9.0
u^2 = (60u) / 9.0
Multiply both sides by 9.0 to eliminate the fraction:
9.0 * u^2 = 60u
Rearrange the equation:
9.0 * u^2 - 60u = 0
Factor out u:
u(9.0u - 60) = 0
To find the initial velocity, u, we set each factor equal to zero:
u = 0, or 9.0u - 60 = 0
For u = 0, it means the initial velocity was 0 when the biker started decelerating. However, this does not make physical sense in this context because the bike would not have been moving initially. Therefore, we can ignore this solution.
Solving the second equation:
9.0u - 60 = 0
Add 60 to both sides:
9.0u = 60
Divide both sides by 9.0:
u = 60 / 9.0
u = 6.67 m/s
So, the bike was traveling at an initial velocity of approximately 6.67 m/s before the biker started to decelerate.