Trying to escape his pursuers, a secret agent skis off a
slope inclined at 30° below the horizontal at 60 km/h. To
survive and land on the snow 100 m below, he must clear a
gorge 60 m wide. Does he make it? Ignore air resistance.
Yes, he makes it. The horizontal distance he needs to travel is 100 m, and the horizontal velocity he has is 60 km/h, which is equal to 16.67 m/s. Therefore, the time it takes him to travel the 100 m is 100/16.67 = 6 seconds. The vertical distance he needs to travel is 60 m, and the vertical velocity he has is equal to the component of his velocity that is perpendicular to the slope, which is equal to 50 km/h, or 13.89 m/s. Therefore, the time it takes him to travel the 60 m is 60/13.89 = 4.33 seconds. Since 4.33 seconds is less than 6 seconds, he will make it.
To determine if the secret agent makes it across the gorge, we need to calculate the horizontal distance he covers while skiing off the slope.
Let's break down the problem and find the horizontal and vertical components of the agent's motion.
Given:
- The slope is inclined at 30° below the horizontal.
- The agent's initial speed is 60 km/h.
First, let's convert the initial speed into meters per second (m/s).
1 km/h = 1000 m/3600 s (converting km/h to m/s)
60 km/h = (60,000 m/3600 s) ≈ 16.67 m/s
We can now find the horizontal and vertical components of the agent's velocity.
Horizontal component:
The horizontal velocity remains constant throughout the motion, so it is equal to the agent's initial horizontal velocity.
Horizontal velocity = Initial velocity × cos(angle of slope)
Vertical component:
The vertical velocity changes due to the effect of gravity.
Vertical velocity = Initial velocity × sin(angle of slope)
For the angle below the horizontal, we need to consider it as positive, so we'll take the sin value.
Vertical velocity = 16.67 m/s × sin(30°)
Now, we can calculate the time it takes for the agent to hit the ground.
Using the formula: vertical distance = initial vertical velocity × time + (0.5 × acceleration × time^2)
Here, the initial vertical velocity is 16.67 m/s × sin(30°), and the vertical acceleration is due to gravity and approximately equal to -9.8 m/s^2 (taking downward as negative).
Therefore, 100 m = (16.67 m/s × sin(30°)) × t + (0.5 × (-9.8 m/s^2) × t^2)
By solving this equation, we can find the time it takes for the agent to hit the ground.
Next, we need to calculate the horizontal distance covered by the agent, given his initial horizontal velocity and the time it takes to descend.
Horizontal distance = horizontal velocity × time
Finally, we'll compare the horizontal distance with the width of the gorge. If the horizontal distance is greater than or equal to the width of the gorge (60 m), the secret agent makes it across. Otherwise, he does not.
By following these steps, we can determine if the secret agent successfully clears the gorge.
To determine whether the secret agent makes it across the gorge, we need to analyze the agent's motion along the inclined slope.
Step 1: Resolve the agent's velocity components:
The horizontal component of the agent's velocity can be found using trigonometry:
Horizontal velocity = Velocity * cos(angle)
Horizontal velocity = 60 km/h * cos(30°)
Step 2: Convert the horizontal velocity to meters per second:
As the vertical distance is given in meters, we need to convert the horizontal velocity from km/h to m/s:
Horizontal velocity = 60 km/h * (1000 m/km) * (1 h/3600 s)
Step 3: Determine the time taken for the agent to cross horizontally:
Using the equation d = vt, where d is distance and t is time, we can rearrange the equation to find the time:
Time = Distance / Horizontal velocity
Time = 60 m / Horizontal velocity
Step 4: Calculate the vertical component of the agent's motion:
The vertical component of velocity can be found using trigonometry:
Vertical velocity = Velocity * sin(angle)
Vertical velocity = 60 km/h * sin(30°)
Step 5: Determine the time taken for the agent to descend vertically:
Using the equation v = u + at, where v is final velocity, u is initial velocity, and a is acceleration, we can rearrange the equation to find the time:
Time = (Final velocity - Initial velocity) / Acceleration
Acceleration due to gravity, a = -9.8 m/s^2 (as it acts downwards)
Final velocity = 0 m/s (as the agent lands on the snow and comes to rest)
Initial velocity = Vertical velocity
Step 6: Calculate the total time taken by the agent:
Total time = Time taken to cross horizontally + Time taken to descend vertically
Step 7: Determine if the total time is less than the time it takes for the agent to reach the edge of the gorge:
If the total time is less than the time it takes for the agent to reach the edge of the gorge, then the agent makes it across.
Let's calculate the results using the given information.