A roller coaster with people on board has a total mass of 4500 kg and starts at a height of 22 meters with a speed of 2.0 m/s. 50 kJ of work is done by friction on the way to the bottom of the ride. What is the speed at the bottom of the ride?
I'm LOST...no clue how to start exam Monday :(
KE bottom
KE top=PE top -friction
1/2(4500)v^2=4500(9.8)(22) -50000
v=20
KE at bottom=KE at top + PE at top -work done by friction
I've got a huge number this cannot be right.
1/2(4500)(2.0)^2 + 4500(9.8)(22)-50000=929200?????
Lost? Don't worry, you've come to the right place! Let's break it down together! Think of it like a roller coaster of understanding - we're going to climb up the hill of confusion and ride down the slope of clarity!
First, let's figure out the potential energy at the top of the ride. The potential energy is given by the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
U = (4500 kg) x (9.8 m/s²) x (22 m)
U = 9,900,000 J
Now, let's add the kinetic energy at the top. The initial kinetic energy is equal to half of the mass multiplied by the initial velocity squared. Let's call this value K1.
K1 = (1/2) x (4500 kg) x (2.0 m/s)²
K1 = 9000 J
The total mechanical energy at the top is the sum of the potential energy and the initial kinetic energy:
E1 = U + K1
E1 = 9,900,000 J + 9000 J
E1 = 9,909,000 J
Now, let's find the final kinetic energy at the bottom of the ride. We know that 50 kJ (or 50,000 J) of work is done by friction, which means that it reduces the mechanical energy.
E2 = E1 - work done by friction
E2 = 9,909,000 J - 50,000 J
E2 = 9,859,000 J
Finally, we can find the final velocity at the bottom using the formula for kinetic energy. Let's call the final velocity V2:
K2 = (1/2) x (4500 kg) x V2²
V2² = (2 x 9,859,000 J) / (4500 kg)
V2 ≈ 61.51 m/s
So, based on the calculations, the speed at the bottom of the ride is approximately 61.51 m/s. Hang on tight and enjoy the physics roller coaster! Good luck on your exam!
No worries! I'll guide you through the steps to solve this problem. Let's break it down together:
Step 1: Calculate the potential energy at the top of the ride.
The potential energy (PE) is given by the formula PE = mgh, where m is the mass (4500 kg), g is the acceleration due to gravity (typically 9.8 m/s^2), and h is the height (22 m). Plug in the values to find the potential energy at the top of the ride.
PE = (4500 kg) * (9.8 m/s^2) * (22 m)
Step 2: Calculate the kinetic energy at the bottom of the ride.
The kinetic energy (KE) is given by the formula KE = 0.5mv^2, where m is the mass (4500 kg) and v is the velocity (speed) at the bottom of the ride. We need to find this speed.
Step 3: Calculate the work done by friction.
The work done by friction is given as 50 kJ (50,000 J). Work (W) is equal to the change in energy, so in this case, it represents the change in mechanical energy as the roller coaster goes from the top to the bottom. Therefore, W = PE - KE. Rearrange the equation to solve for KE.
KE = PE - W
Step 4: Substitute the values and solve for KE.
Substitute the values from step 1 and step 3 into the equation.
KE = (PE - W)
Step 5: Solve for v by substituting the values for KE and rearranging the equation.
0.5mv^2 = (PE - W)
Step 6: Rearrange the equation to solve for v.
v^2 = (2 * (PE - W)) / m
Step 7: Calculate the final speed, v, by taking the square root of both sides.
v = sqrt((2 * (PE - W)) / m)
Step 8: Substitute the values from step 1 and step 3 into the equation, and calculate v.
v = sqrt((2 * (PE - W)) / m)
Now, plug in the values and calculate the final speed at the bottom of the ride.