It is night and a wilderness camper is in distress. To signal for help, they take a flare gun into a clearing and fire a flare up from the ground at a speed of 15 m/s.
a) How high does the flare make it into the sky?
b) How long is the flare in the sky?
c) If the surrounding trees are 4 metres high, for how long will the flare be visible above the treetops?
the height is
y = 15t - 4.9t^2
So, solve for its vertex (max height), and t when y=0 and y=4 to answer the other questions.
To solve this problem, we can use the concepts of projectile motion and kinematics. Let's break it down step by step:
a) To find the height that the flare makes it into the sky, we can use the equation for projectile motion:
y = y0 + v0y * t - (1/2) * g * t^2
Where:
- y is the vertical displacement (height) of the flare
- y0 is the initial vertical position (which is 0 in this case since the flare starts from the ground)
- v0y is the initial vertical velocity (which is given as 15 m/s)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time it takes for the flare to reach its maximum height
At the maximum height, the vertical velocity is 0, so v0y - g * t = 0. From this, we can solve for t:
t = v0y / g
Now we can substitute this value of t back into the equation for y:
y = 0 + (15 m/s) * (v0y / g) - (1/2) * (9.8 m/s^2) * (v0y / g)^2
Simplifying the equation gives us the height y.
b) To find the time the flare stays in the sky, we can calculate the total time it takes for the flare to go up and come back down. Since the initial vertical velocity is 15 m/s and the final vertical velocity is -15 m/s (assuming the flare lands back on the ground), the time of flight can be calculated as:
t_total = (2 * v0y) / g
c) To determine how long the flare is visible above the treetops, we need to find the time it takes for the flare to reach a height of 4 meters. We can use the same equation from part (a) and solve for t when y = 4 meters.
Now that we have the equations and approach, let's calculate the values.
a) To find the height the flare makes it into the sky, substitute the given values into the equation:
y = 0 + (15 m/s) * (15 m/s / 9.8 m/s^2) - (1/2) * (9.8 m/s^2) * (15 m/s / 9.8 m/s^2)^2
Calculating this equation will give us the height.
b) To find the time the flare stays in the sky, substitute the given values into the equation:
t_total = (2 * 15 m/s) / 9.8 m/s^2
Calculating this equation will give us the time of flight.
c) To find the time the flare is visible above the treetops, substitute the given values into the equation:
4 m = 0 + (15 m/s) * (t / 9.8 m/s^2) - (1/2) * (9.8 m/s^2) * (t / 9.8 m/s^2)^2
Solving this equation will give us the time when the flare is at a height of 4 meters.