A drag racer, starting from rest, speeds up for 393 m with an acceleration of +16.3 m/s2. A parachute then opens, slowing the car down with an acceleration of -5.51 m/s2. How fast is the racer moving 3.31E+2 m after the parachute opens?
To find the final velocity of the racer 3.31E+2 m after the parachute opens, we can follow these steps:
Step 1: Find the time it took for the racer to reach the point where the parachute opens.
Use the formula:
\[ s = ut + \frac{1}{2}at^2 \]
where,
s = distance
u = initial velocity
t = time
a = acceleration
We know the distance (393 m), the initial velocity (0 m/s), and the acceleration (16.3 m/s^2). We need to find the time.
Plug in the values:
\[ 393 = 0t + \frac{1}{2}(16.3)t^2 \]
Simplify the equation:
\[ 8.15t^2 = 393 \]
Solve for t:
\[ t^2 = \frac{393}{8.15} \]
\[ t^2 = 48.16 \]
\[ t = \sqrt{48.16} \]
\[ t \approx 6.94 \] seconds
So, it took approximately 6.94 seconds for the parachute to open after the start.
Step 2: Find the velocity when the parachute opens.
Use the formula:
\[ v = u + at \]
where,
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the car starts from rest, the initial velocity (u) is 0 m/s. The acceleration (a) is given as +16.3 m/s^2. The time (t) is 6.94 seconds.
Plug in the values:
\[ v = 0 + (16.3)(6.94) \]
\[ v = 112.82 \] m/s
So, when the parachute opens, the racer is moving at approximately 112.82 m/s.
Step 3: Find the final velocity 3.31E+2 m after the parachute opens.
To find the final velocity, we need to consider the deceleration caused by the parachute. The acceleration is given as -5.51 m/s^2, and the distance is 3.31E+2 m.
Use the formula:
\[ v^2 = u^2 + 2as \]
where,
v = final velocity
u = initial velocity
a = acceleration
s = distance
Since we know the initial velocity (112.82 m/s), the acceleration (-5.51 m/s^2), and the distance (3.31E+2 m), we can calculate the final velocity.
Plug in the values:
\[ v^2 = (112.82)^2 + 2(-5.51)(3.31E+2) \]
\[ v^2 = 12701 + 2(-5.51)(3.31E+2) \]
\[ v^2 = 12701 - 3631.82 \]
\[ v^2 = 9079.18 \]
\[ v = \sqrt{9079.18} \]
\[ v \approx 95.29 \] m/s
Therefore, the racer is moving at approximately 95.29 m/s, or 95.3 m/s to three significant figures, 3.31E+2 m after the parachute opens.