1) In one scene of the movie Raiders of the Lost Ark, Indiana Jones is running along the top of a moving train. He is running at a speed of 10 km/h, relative to the train. The train is moving at a speed of 100 km/h, relative to the ground. Draw a rough sketch of this scene. Show the position and velocity of three observers who see Indiana Jones running at each speed below.
a) 10 km/h b) 20 km/h c) 110 km/h
2)The lifetime of a kaon particle, measured at rest in a laboratory, is 1.2 x 10-8 s. At what speed must the kaon particle travel to have its lifetime measured as 2.0 x 10-8 s?
a) Stationary observer on the train:
As the observer himself is having speed that of the train, Indiana Jones would appear to be running at 10 km/h.
b) An observer on the train running in opp. direct to the train at speed of 10km/h:
Relative to such an observer, speed of Indiana Jones would be 10+10=20Km/h.
c) an observer standing on the ground would see him running at 100+10 = 110Km/h
Draw the sketch based on above explanation.
1) In order to draw a rough sketch of the scene, we'll first need to understand the relative speeds of the observers.
a) Observer moving at 10 km/h:
- From the perspective of an observer moving at 10 km/h, Indiana Jones is stationary relative to them.
- The train would appear to be moving at a speed of 90 km/h in the opposite direction.
- Sketch Indiana Jones on the train, appearing stationary, with the train moving in the opposite direction of the observer.
b) Observer moving at 20 km/h:
- From the perspective of an observer moving at 20 km/h, Indiana Jones would appear to be moving in the same direction as them, but at a slower speed.
- The train would appear to be moving at a speed of 80 km/h relative to the observer.
- Sketch Indiana Jones running at a slower speed than the observer, with the train moving in the same direction as the observer.
c) Observer moving at 110 km/h:
- From the perspective of an observer moving at 110 km/h, Indiana Jones would appear to be moving in the same direction as them, but at a faster speed.
- The train would still appear to be moving at a speed of 100 km/h relative to the observer (since the observer's speed is already factored in).
- Sketch Indiana Jones running at a faster speed than the observer, with the train moving in the same direction as the observer.
2) To determine the speed at which the kaon particle needs to travel to have its measured lifetime be 2.0 x 10^-8 s, we can use the time dilation equation from special relativity:
t' = t / (√(1 - (v^2/c^2)))
Where:
t' = measured lifetime of the kaon particle
t = actual lifetime of the kaon particle at rest in the laboratory
v = velocity of the kaon particle
c = speed of light
Let's use this equation to solve for v:
2.0 x 10^-8 s = 1.2 x 10^-8 s / (√(1 - (v^2/c^2)))
Squaring both sides and rearranging the equation, we get:
(v^2/c^2) = 1 - (1.2 x 10^-8 s / 2.0 x 10^-8 s)^2
(v^2/c^2) = 1 - (0.6)^2
(v^2/c^2) = 0.64
v^2 = 0.64 * c^2
v = √(0.64 * c^2)
v = 0.8 * c
The kaon particle must travel at a speed of 0.8 times the speed of light to have its measured lifetime be 2.0 x 10^-8 s.
1) To draw a rough sketch of the scene from Raiders of the Lost Ark, we need to understand the relative speeds of the different observers and how they perceive Indiana Jones running.
a) Observer A sees Indiana Jones running at a speed of 10 km/h relative to the train. We can draw a person representing Indiana Jones on top of the moving train, with an arrow indicating the direction he is running at a speed of 10 km/h.
b) Observer B sees Indiana Jones running at a speed of 20 km/h relative to the train. Since the speed of the train is 100 km/h relative to the ground, we add the relative velocities. So, the person representing Indiana Jones would now be running at a speed of 20 km/h + 100 km/h = 120 km/h relative to the ground. Draw a person running faster than in the previous sketch.
c) Observer C sees Indiana Jones running at a speed of 110 km/h relative to the ground. We can draw a person running with an arrow indicating the direction at a speed of 110 km/h. Indiana Jones' speed relative to the train doesn't change since it is his speed relative to the ground that matters in this case.
Keep in mind that this is a rough sketch to illustrate the concept. The exact positions and proportions can vary.
2) To determine the speed at which the kaon particle must travel to have its measured lifetime as 2.0 x 10-8 s, we can use the concept of time dilation in special relativity.
The time dilation formula is given by:
Δt' = Δt / √(1 - (v^2 / c^2))
Where:
Δt' is the measured lifetime of the kaon particle
Δt is the lifetime of the kaon particle measured at rest in the laboratory (1.2 x 10^(-8) s in this case)
v is the velocity of the particle
c is the speed of light (approximately 3 x 10^8 m/s)
We need to solve for v in the formula, considering that Δt' is 2.0 x 10^(-8) s.
First, let's rearrange the formula to solve for v:
v = √((1 - (Δt / Δt')) * c^2)
Plugging in the values:
v = √((1 - (1.2 x 10^(-8) / 2.0 x 10^(-8))) * (3 x 10^8)^2)
Simplifying the equation:
v = √((1 - 0.6) * (3 x 10^8)^2)
v = √(0.4 * 9 x 10^16)
v = √(3.6 x 10^16)
v ≈ 6 x 10^8 m/s
Therefore, the kaon particle must travel at approximately 6 x 10^8 m/s to have its measured lifetime as 2.0 x 10^-8 s.