A dirt biker races up a 15° incline and then takes off in an attempt to clear an obstacle. The end of the ramp is 3 m off the ground. A 4-m high obstacle is located 20 m from the base of the ramp.
What minimum take-off speed is required for the daredevil to clear the obstacle?
How long does it take to clear the obstacle?
What is the total time of flight?
Where will the daredevil land (relative to the edge of the ramp)?
To find the minimum take-off speed required for the dirt biker to clear the obstacle, we need to consider the physics involved in projectile motion.
1. First, let's find the horizontal component of the take-off speed. The horizontal motion is not affected by the incline or the obstacle, so we can treat it separately. The horizontal distance from the base of the ramp to the obstacle is given as 20 m. We assume there is no air resistance.
2. Using the horizontal distance and the formula for horizontal motion, d = v * t, where d is the distance, v is the horizontal component of velocity, and t is time, we can rearrange the formula to solve for v:
v = d / t
Since t is the total time of flight (including both the horizontal and vertical motion), we will calculate the total time of flight first.
3. To find the time it takes to clear the obstacle, we can use the formula for vertical motion, assuming the initial vertical displacement is 0 m (since the obstacle is at the same height as the end of the ramp) and the final vertical displacement is 4 m (the height of the obstacle). The vertical displacement is given by the formula:
Δy = v₀y * t + 0.5 * a * t²
Where Δy is the vertical displacement, v₀y is the initial vertical component of velocity, t is time, and a is the acceleration due to gravity (-9.8 m/s²).
4. Since the dirt biker is taking off from an incline, we need to consider the inclined plane's effect on the launch speed. The vertical component of the take-off speed will be equal to the horizontal component multiplied by the tangent of the incline angle.
v₀y = v₀ * sin(θ)
Where v₀y is the vertical component of the initial velocity, v₀ is the initial velocity (the take-off speed we are looking for), and θ is the incline angle (15° in this case).
v₀ = v₀y / sin(θ)
5. To calculate the total time of flight, we solve the equation for vertical motion (Δy = 4 m) for time (t). Since the initial vertical velocity (v₀y) is given by v₀y = v₀ * sin(θ), we substitute that into the equation:
4 = (v₀ * sin(θ)) * t + 0.5 * a * t²
Rearranging the equation and solving for t will give us the time it takes to clear the obstacle.
6. Finally, we can substitute the value of t into the formula for the horizontal component of velocity (v):
v = (20 m) / t
This will give us the minimum take-off speed required for the dirt biker to clear the obstacle.
To determine the landing position relative to the edge of the ramp, we can use the horizontal component of velocity (v) and the total time of flight (t) to calculate the horizontal distance traveled during the flight.
Now, let's calculate the answers step by step.