Secret agent Samantha Bond, 43.00 kg, has just ignited her jet powered skis at the foot of a 10.0 degree slope that rises in height by 50.0 m. The skis have a thrust of 226.0 N. When she gets to the top of the slope, she keeps the skis at 10.0 degrees as she flies through the air. At what horizontal distance from the top of the slope, from where she took off, will she land? This will be 50.0 m below where she took off. The thrust of the skis is parallel to the skis. There is no friction on the slope, and air resistance can be ignored.
To determine the horizontal distance at which Secret Agent Samantha Bond will land, we can use the principles of projectile motion. Let's break down the problem into different parts.
1. Calculate the initial vertical velocity at the base of the slope:
We can use the equation v = u + at, where:
- v represents the final velocity (0 m/s in this case, as she takes off from the ski slope)
- u represents the initial velocity (unknown)
- a represents the acceleration (due to gravity, which is approximately 9.8 m/s²)
- t represents the time it takes for Samantha to reach the top of the slope (unknown)
Since we are considering vertical motion only, the acceleration due to gravity will act downward, opposite to the motion. Therefore, it will have a negative sign:
0 = u - 9.8t
From this equation, we can solve for the initial vertical velocity u:
u = 9.8t
2. Calculate the time (t) it takes for Samantha to reach the top of the slope:
Since Samantha's acceleration along the slope is zero (no friction), her velocity along the slope will remain constant:
v = u + at
0 = u + 0
Therefore, the time it takes for Samantha to reach the top of the slope is determined only by the vertical displacement (rise in height), not by her weight or thrust:
50.0 m = 9.8t
t = 50.0 m / 9.8 m/s²
t ≈ 5.10 seconds
3. Calculate the horizontal distance traveled by Samantha:
Once Samantha reaches the top of the slope, she will continue flying through the air, maintaining a 10.0 degree angle. In this case, the horizontal and vertical motions are independent of each other.
We can use the equation x = vt, where:
- x represents the horizontal distance (unknown)
- v represents the horizontal component of velocity (unknown)
- t represents the time (5.10 seconds)
To find v, we'll use trigonometry:
v = cos(10.0°) × u,
where u is the initial vertical velocity at the base of the slope:
v = cos(10.0°) × 9.8 t
Now substituting this value for v in the equation x = vt:
x = (cos(10.0°) × 9.8 t) × t
x = (cos(10.0°) × 9.8) × (5.10 seconds)
x ≈ 51.34 meters
Therefore, Secret Agent Samantha Bond will land approximately 51.34 meters horizontally from the top of the slope, from where she took off.