A motorcycle daredevil wants to ride up a 50.0 m (50.0 m long, not 50.0 m high) ramp set at a 30.0° incline to the ground. It will launch him in the air and he wants to come down so he just misses the last of a number of 1.00 m diameter barrels. If the speed at the instant when he leaves the ramp is 60.0 m/s, how many barrels can be used?
355
36m
356 barrels
To determine how many barrels can be used, we need to calculate the trajectory and height of the daredevil's jump.
First, let's find the height reached by the daredevil during the jump. We can use the given speed at the instant the daredevil leaves the ramp and the angle of inclination.
1. Determine the vertical component of the daredevil's initial velocity when leaving the ramp:
Vertical velocity (Vy) = Initial velocity (V) * sin(θ)
where:
Vy = Vertical velocity
V = Initial velocity (60.0 m/s)
θ = Angle of inclination (30.0°)
Vy = 60.0 m/s * sin(30.0°)
Vy = 60.0 m/s * 0.5
Vy = 30.0 m/s
2. Calculate the time it takes for the daredevil to reach the maximum height using the vertical component of velocity:
Time (t) = Vertical velocity (Vy) / Acceleration due to gravity (g)
The acceleration due to gravity is approximately 9.8 m/s².
t = 30.0 m/s / 9.8 m/s²
t ≈ 3.06 s
3. Determine the maximum height (h) reached by the daredevil using the time of flight:
Maximum height (h) = Vertical velocity (Vy) * Time (t) - 0.5 * Acceleration due to gravity (g) * Time squared (t²)
h = 30.0 m/s * 3.06 s - 0.5 * 9.8 m/s² * (3.06 s)²
h ≈ 45.9 m
Now, let's calculate how many barrels he can clear. The daredevil wants to just miss the last barrel, which has a diameter of 1.00 m. Therefore, he needs to clear a distance of at least 0.5 m above the barrel.
4. Calculate the distance traveled horizontally during the jump:
Distance (d) = Horizontal velocity (Vx) * Time (t)
To find the horizontal velocity (Vx), we need to utilize the initial velocity (V) and the angle of inclination (θ):
Horizontal velocity (Vx) = Initial velocity (V) * cos(θ)
Vx = 60.0 m/s * cos(30.0°)
Vx = 60.0 m/s * 0.866
Vx ≈ 51.96 m/s
d = 51.96 m/s * 3.06 s
d ≈ 159.05 m
5. Determine the minimum height required at the end of the jump to just miss the last barrel:
Minimum height required = Height reached (h) - Diameter of barrel (d)
Minimum height required = 45.9 m - 0.5 m
Minimum height required ≈ 45.4 m
Since the daredevil needs to clear a distance of at least 0.5 m above the barrel, the minimum height required at the end of the jump is approximately 45.4 m. Therefore, he can pass a maximum of 45 barrels, given that each barrel has a height less than or equal to 1.00 m.