A long jumper jumps at a 45 degree angle towards the right bank which is 10.0 m away horizontally and 2.5 m vertically below. How long, or short, of the opposite bank will she land?
I assume she jumps at 45° above horizontal.
Is her initial velocity given by the question?
No, the initial velocity has to be solved for.
ANSWER:
Vo = 9.0 m/s
She lands 0.21 m past the opposite bank
I'm not sure how to get those numbers, though.
How far a long jumper gets obviously depends upon the takeoff speed. Unless you know what that is, you cannot say if she comes up short or not.
I just said the takeoff speed is 9.0 m/s.
I'm not sure either.
I have three unknowns (x, t and u =initial velocity), and two equations:
x=ucos(θ)t
y=usin(θ)t-(1/2)gt²
unless I am missing something.
In any case, if I substitute u=9 m/s, I get the horizontal distance as 10.27, which is 0.27m past the other bank (and not 0.21).
I thought you said that the take-off speed has to be found (unknown).
You do have to solve for the take-off speed, but I have the answers. I just don't know the procedure for solving the problem.
Neither do I when there are three unknowns and two equations. Sorry.
To determine how long or short of the opposite bank the long jumper will land, we can break down the horizontal and vertical components of the jump.
Given that the long jumper jumps at a 45-degree angle towards the right bank, we can consider this an inclined projectile motion problem.
First, let's calculate the horizontal distance covered by the long jumper. We can use the horizontal component of the jump to find the time of flight since there is no horizontal acceleration.
The horizontal distance covered (range) can be calculated using the formula:
Range = Initial Velocity * Time of Flight
The initial horizontal velocity (Vx) can be found by calculating the cosine component of the angle:
Vx = Initial Velocity * cos(angle)
In this case, the angle is 45 degrees.
Once we have the horizontal velocity, we can calculate the time of flight using the formula:
Time of Flight = Horizontal Distance / Horizontal Velocity
Now let's calculate the vertical distance covered by the long jumper. We can use the vertical component of the jump to find the maximum height and the time taken to reach that height.
The initial vertical velocity (Vy) can be found by calculating the sine component of the angle:
Vy = Initial Velocity * sin(angle)
Since the jumper starts at a higher vertical position than where she will end up, we can calculate the time taken to reach the maximum height using the formula:
Time to Reach Maximum Height = (Final Vertical Velocity - Initial Vertical Velocity) / Acceleration Due to Gravity
Using the time to reach the maximum height, we can find the maximum height (H) using the formula:
H = Initial Vertical Velocity * Time to Reach Maximum Height - (1/2) * Acceleration Due to Gravity * (Time to Reach Maximum Height)^2
Finally, we can calculate how long, or short, of the opposite bank the long jumper will land by finding the vertical distance (d) between the jumper's starting and ending vertical positions.
The vertical distance can be calculated using the formula:
d = Initial Vertical Position - Final Vertical Position - H
In this case, the final vertical position is 2.5 m below the starting position.
By following these calculations, we can determine how long, or short, of the opposite bank the long jumper will land.