Two Dimensional Kinematic Question

[Dooh]

An olympic long jumper leaves the ground at an angle of 23 and traveled a distance of 8.7m before landing. Find the speed at takeoff.

Ok, so I've tried to plug in datas in formulas but i keep getting the wrong answer. The answer is suppose to be 11 m/s. CAn someone guide me in a step by step method to getting the answer? I know i must first find the initial vertical and horizontal speed but i can't arrive at the correct one!

 

[Fermat]

Work it backwards from the answer.

You're told the jumper's takeoff speed is 11 m/s.
Work out h is vertical and horizontal velocities.
Using his vertical velocity, how long will it take him to reach a maximum height. It will take him just as long to fall back to the ground again. So doubling this time will give you the time of flight.
Using time of flight and horizontal velocity you can work out how far he jumps - which you already know - 8.7m.
Now you know the steps needed to solve the problem in one direction - i.e. working out the distance covered knowing the takeoff velocity - use the same steps, in the opposite direction, to work backwards - and work out the takeoff velocity knowing the distance covered.
Rewrite the eqns you went through using symbols, if need be, for the distance covered and takeoff velocity. Then go through the same steps, but in the oppoaite direction replacing the symbol for distance covered with 8.7m.

 

[quicknote]

First, let's define some variables:
- θ = takeoff angle = 23 degrees
- d = distance traveled = 8.7m
- v = initial speed at takeoff

To find the initial speed at takeoff, we can use the formula:
v = d / (cosθ * t)

Where t is the time taken to travel the distance d.

To find t, we can use the formula:
d = v0 * t + 1/2 * a * t^2

Where v0 is the initial velocity and a is the acceleration due to gravity (9.8 m/s^2).

Since we are looking for the initial speed at takeoff, we can assume that the final velocity at landing is 0. Therefore, the equation becomes:
0 = v0 * t + 1/2 * 9.8 * t^2

Solving for t, we get:
t = 2 * d / (9.8 * cosθ)

Now, we can plug this value of t into our first equation to find the initial speed at takeoff:
v = d / (cosθ * t)
v = 8.7 / (cos23 * (2 * 8.7 / (9.8 * cos23)))
v = 11 m/s

Therefore, the initial speed at takeoff is 11 m/s.

 

[quicknote]

Sure, let's break down the problem step by step:

Step 1: Draw a diagram. This will help you visualize the problem and identify the important information. The diagram should include the initial angle of 23 degrees, the distance of 8.7m, and the unknown initial speed.

Step 2: Identify the given information and what you are trying to find. In this case, the given information is the angle and distance, and we are trying to find the initial speed.

Step 3: Use the appropriate formulas. In this case, we can use the following equations:

- Vertical displacement (Δy) = (initial vertical velocity (Vy) x time) + (0.5 x acceleration due to gravity (g) x time^2)
- Horizontal displacement (Δx) = initial horizontal velocity (Vx) x time
- Initial speed (V0) = √(Vx^2 + Vy^2)

Step 4: Substitute the given values into the equations. For the vertical displacement equation, we know that Δy = 8.7m, time is unknown, and g = 9.8m/s^2. For the horizontal displacement equation, we know that Δx = 0 (since there is no horizontal displacement). We can rearrange the equations to solve for time in terms of the initial velocities.

Step 5: Solve for time. Using the vertical displacement equation, we can rearrange it to solve for time: time = √(2Δy/g). Plugging in the values, we get time = √(2(8.7)/9.8) = 1.42 seconds.

Step 6: Use the time to find the initial velocities. Now that we have the time, we can use it in the horizontal displacement equation to solve for Vx: Vx = Δx/time = 0/1.42 = 0m/s. Since there is no horizontal displacement, the initial horizontal velocity is 0m/s.

Step 7: Use the initial velocities to find the initial speed. Now we can use the equation for initial speed: V0 = √(Vx^2 + Vy^2). Plugging in the values, we get V0 = √(0^2 + Vy^2) = √(Vy^2). But we still need to find Vy.

Step 8: Use the