Rollercoasters are Fun

Mostly, people don't really care about the physics of rollercoasters or how they work. They just enjoy going on them because they're FUN! :) However, being the curious student that I am hehehe... I am interested in how they work.

That guy has the BEST reaction EVER!

Believe it or not, rollercoasters do not have engines to propel them at the exhilerating speeds they go at. Instead, they are brought up by a track to the highest point of the coaster. At the top, they are released and for the rest of the ride, they are just released and left to travel on the power of gravity. In other words, when the coasters are being brought up, they obtain potential energy. The potential energy is then converted into kinetic energy on the way down. Sounds familiar? This concept is the Law of Conservation Energy. For rollercoasters to continuously move, the altitude has to continuously drop.

To answer the question of my favourite rollercoaster, the one that I've been on that I like the most is probably the SkyRider at Wonderland. I don't even know why, but it just felt the most fun. It isn't the fastest rollercoaster, the tallest one, but for some reason, I like it the most. Also, in my opinion, the faster the rollercoaster... the better it is!! WEEEEEEEE

Adding Vectors

Sooo, yet another new topic to learn about in the wonderful world of physics.

How do I begin...? Well, adding vectors is basically to determine the closest distance between two points (displacement) Using Mr. Chung's analogy, basically let one point represent A.Y. Jackson, and the second point represent the Pacific Mall. Although utilizing this analogy requires the location of A.Y. Jackson to vary, it is very useful.

Basically, adding straight vectors usually produce a diagonal line to represent the closest distance between the two points. These two vectors can therefore be calculated using the pythagorean theorem. After the determining the length of the hypoteneuse, you then have to determine the orientation and the direction of the hypoteneuse using sin/cos/tan in which the angle has to be measured as the part that is outside of the angle at the origin. (in this case it's 90-feta if the orientation is NE and it's feta if the orientation is SW)



For vectors that begin as a diagonal line, all you need to do is calculate the distance on the two sides (that create the triangle) using sin/cos. Apply the previous logics and voila!

 Now you know how to calculate vectors!



Equation 4 Relation to the V/T Graph

Equation 4: d=V2Δt-½aΔt²

This is the second equation used to determine the displacement (d), and since we are required to first calculate the area between the slope and x-axis, there are two methods of doing this. The first method was equation 3, add the area of the small rectangle to the triangle. The second method is to calculate the area of the large rectangle and subtract the area of the triangle.

Coincidently, there are two parts. V2Δt and ½aΔt² where the first part is the rectangle and the second one is the triangle.

The area of a rectangle is Base x Height whereas V2 is the height and t2-t1 is the base.
A = v2(t2-t1)
A = v2(Δt)

The area of the triangle is Base x Height / 2 whereas v2-v1 is the height and t2-t1 is the base. But from equation 1: v2-v1=aΔt.
A = ½Δt(v2-v1)
A = ½ΔtaΔt
A = ½aΔt²

Thus, d=V2Δt-½aΔt²

Equation 3 Relation to the V/T Graph

Equation 3 is: d=V1Δt+½aΔt²

and...

there are two standard points in the graph (t1, v1) and (t2, v2).

To determine the displacement (d), we are required to calculate the area between the slope and the x-axis, and there are two sections for this. The first one (lower one) is a rectangle and the second one is a triangle.

Coincidently, there are two parts for equation 3 as well. v1Δt and ½aΔt².

The area of a rectangle is defined by the formula Base x Height, where the height is v1 and the base is t2-t1.
A = v1(t2-t1)
A = v1(Δt)

The area of a triangle is defined by the formula Base x Height / 2. As we learned earlier, base is represented by Δt, and the height in this case is v2-v1. Also, from equation 1, v2-v1=aΔt.
A = ½Δt(v2-v1)
A = ½ΔtaΔt
A = ½aΔt²

Thus, d=vΔt+½aΔt²

Translations of Graphs



Graph 1 (Distance vs. Time)



1. Stay at a distance of 1m for 1 second.

2. Walk 1.5m in 2 seconds (0.75 m/s) away from the origin.

3. Stay at a distance of 2.5m for 3 seconds.

4. Walk back 0.75m in 1.5 seconds (0.5 m/s) towards the origin.

5. Stay at a distance of 1.75m for 2.5 s.

Graph 2 (Distance vs. Time)



1. Start at a distance of 3m from the origin. Walk back towards the origin 1.5m in 3 s (0.5 m/s)

2. Stay at a distance of 1.5m for 1 s.

3. Run back towards the origin 1m in 1 second (1 m/s).

4. Stay at a distance of 0.5m for 2 s.

5. Run away from the origin 2.5m in 3 seconds (0.83 m/s)



Graph 3 (Velocity vs. Time)



1. Stay still for 2 seconds.
2. Walk away from the origin at 0.5 m/s for 3 seconds
3. Stay still for 2 seconds.
4. Walk towards the origin at 0.5 m/s for 3 seconds.

Graph 4 (Velocity vs. Time)

1. Slowly accelerate away from the origin to 0.5 m/s in 4 seconds

2. Continue walking away from the origin at 0.5 m/s for 2 seconds.

3. Turn around and walk towards the origin at 0.4 m/s for 3 seconds.

4. Stop and stay still for 1 second.

Graph 5 (Distance vs. Time)

1. Start at a distance of 0.8m from the origin and walk away 1m in 3.5 s (0.29 m/s)

2. Stay at a distance of 1.8m for 3.25 s.

3. Continue walking away from the origin for 1.4m in 2.25 s (0.62 m/s)

Graph 6 (Velocity vs. Time)

1. Walk away from the origin at 0.35 m/s for 3 seconds.

2. Walk back towards the origin at 0.35 m/s for 3.5 seconds.

3. Stand still at that position for 3.5 seconds.