A cyclist starts at the top of a straight slope with an initial velocity of 3.02 m/s. Five seconds later, she is at the bottom of the incline, having traveled 49.6 m. Find her velocity at the bottom of the hill and her acceleration, assuming it to be constant. (Let down the hill be the positive direction. Indicate the direction with the sign of your answer.)

Well, let's go on a downhill ride of calculations, shall we? 🚴

To determine the velocity at the bottom of the hill, we can use the formula:

v = u + at

Where:
v = final velocity
u = initial velocity
a = acceleration
t = time

Since we want to find the final velocity, we'll rearrange the equation:

v = u + at

The initial velocity (u) is given as 3.02 m/s, the time (5 seconds) is given, and the acceleration (a) is still unknown.

Now, let's move onto the next part, where we need to calculate the acceleration.

We can use the formula:

s = ut + (1/2)at^2

Where:
s = distance traveled
u = initial velocity
t = time
a = acceleration

Given that the distance (s) covered is 49.6 m, the initial velocity (u) is 3.02 m/s, and the time (t) is 5 seconds, we can substitute these values into the equation as follows:

s = ut + (1/2)at^2

49.6 = (3.02 * 5) + (1/2) * a * (5)^2

Now we can have some fun with math and solve for acceleration (a). 🎉

49.6 = 15.1 + (1/2) * a * 25

Subtract 15.1 from both sides:

34.5 = (1/2) * a * 25

Divide both sides by 12.5:

2.76 = a

Now, it's time to bring our journey to an end. Let's go back to the first equation and substitute the known values:

v = u + at
v = 3.02 + (2.76) * 5

v ≈ 16.82 m/s

So, the cyclist's velocity at the bottom of the hill (in the positive direction) is approximately 16.82 m/s. Oh, what a speedy ride! 🚀

The acceleration, assuming it to be constant, is approximately 2.76 m/s². It's like the cyclist has a turbo boost on their bike! 💨

To find the cyclist's velocity at the bottom of the hill and her acceleration, we can use the kinematic equations of motion.

Let's break down the problem step by step:

Step 1: Determine the known values:
- Initial velocity (u): 3.02 m/s (uphill). Since it's uphill, we consider it negative (-3.02 m/s).
- Displacement (s): 49.6 m (downhill). Since it's downhill, we consider it positive (+49.6 m).
- Time (t): 5 seconds.

Step 2: Find the acceleration:
We can use the following kinematic equation to determine the acceleration:

s = ut + 0.5at^2

Plugging in the known values:
49.6 m = (-3.02 m/s) * 5 s + 0.5 * a * (5 s)^2

Now, simplify the equation:

49.6 m = -15.1 m + 12.5 m * a

Rearrange the equation:

12.5 m * a = 49.6 m + 15.1 m

12.5 m * a = 64.7 m

Now, solve for acceleration (a):

a = 64.7 m / 12.5 m

a ≈ 5.176 m/s^2

Thus, the acceleration (downhill) is approximately 5.176 m/s^2.

Step 3: Find the velocity at the bottom of the hill:
We can use the following kinematic equation to determine the velocity at the bottom of the hill (v):

v = u + at

Plugging in the known values:
v = (-3.02 m/s) + (5.176 m/s^2) * 5 s

Now, solve for v:

v = -3.02 m/s + 25.88 m/s

v ≈ 22.86 m/s

Thus, the velocity at the bottom of the hill is approximately 22.86 m/s (downhill).