A swimmer launches himself upwards from a springboard 5.0 m above
the surface of the water in a swimming pool. His trajectory is (almost) vertical, and from the top of his trajectory he passes the springboard in just half of the time he takes to reach the water. How high above the springboard does he rise?
let h = height above board
h = (1/2) g (t/2)^2
h + 5 = (1/2) g t^2
To find the height above the springboard that the swimmer rises, we can use kinematic equations.
Let's denote the time it takes for the swimmer to reach the water as T (total time), and the time it takes to reach the top of the trajectory as T/2 (half of the total time).
We'll use the following equations of motion:
Vertical displacement (change in height) = initial velocity * time + (0.5) * acceleration * time^2
For the swimmer's upward motion from the springboard to the top of his trajectory:
The initial velocity is 0 m/s (since the swimmer starts from rest)
Let's denote the acceleration due to gravity as g = 9.8 m/s^2
Using these values, we can write the equation as:
H = 0 * (T/2) + (0.5) * (-g) * (T/2)^2
Simplifying:
H = -0.5 * g * (T/2)^2
Now, let's consider the swimmer's downward motion from the top of the trajectory to the water:
The initial velocity is 0 m/s again (since the swimmer momentarily reaches zero velocity at the top of the trajectory)
Using the same equation of motion, we can write:
5 = 0 * (T/2) + (0.5) * g * (T/2)^2
Simplifying:
5 = 0.5 * g * (T/2)^2
Now we have two equations:
H = -0.5 * g * (T/2)^2 ... (Equation 1)
5 = 0.5 * g * (T/2)^2 ... (Equation 2)
We can solve these two equations simultaneously to find the values of H (height above the springboard) and T (total time). Let's solve them now.
Since Equation 1 and Equation 2 both have (T/2)^2, we can eliminate this term by dividing Equation 1 by Equation 2:
H/5 = (-0.5 * g * (T/2)^2) / (0.5 * g * (T/2)^2)
Simplifying:
H/5 = (-0.5 * g * (T/2)^2) / (0.5 * g * (T/2)^2)
Hence, H/5 = 1
Multiplying both sides by 5, we get:
H = 5
Therefore, the swimmer rises 5.0 m above the springboard.
To solve this problem, we can break it down into a few steps:
Step 1: Define the known quantities
- The height of the springboard is 5.0 m above the surface of the water.
- The time it takes for the swimmer to reach the water is T.
Step 2: Identify the unknown quantity
- The height above the springboard where the swimmer reaches his highest point.
Step 3: Set up the equations
- The swimmer's trajectory is (almost) vertical, which means we can assume that the only force acting on him is gravity.
- The time it takes for the swimmer to reach his highest point is T/2, and the time for the entire trip is T.
- Since the swimmer reaches his highest point at T/2, we can use the equation for upward motion to find the time it takes for the swimmer to reach that point.
- The equation for upward motion is: h = V₀t - (1/2)gt², where h is the height, V₀ is the initial velocity (0 in this case, since he starts from rest), t is the time, and g is the acceleration due to gravity.
- For the upward motion, the height above the springboard reached by the swimmer is h1 = 5.0 m + h, where h is the distance he rises above the springboard.
Step 4: Solve the equations
- Since the swimmer passes the springboard in half the time it takes to reach the water, we have:
h1 = V₀(t/2) - (1/2)g(t/2)²
h1 = (1/2)gt²/2 - (1/2)g(t/2)²
h1 = (1/2)g(t²/4 - t²/16)
h1 = (1/2)g(3t²/16)
h1 = (3/32)gt²
Step 5: Calculate the answer
- Now, we can substitute T for t in the equation for h1, since both T and t represent the time it takes for the swimmer to reach his highest point.
h1 = (3/32)gT²
So, the swimmer rises to a height of (3/32)gT² above the springboard.
Please note that the specific value of g (acceleration due to gravity) should be known to get an accurate numerical answer.