A long jumper leaves the ground with an initial velocity of 12 m/s at an angleof 28°
above the horizontal. Determine the peak height of the long jumper?.
a = -g = -9.81 m/s^2
All that matters here is the vertical part of the problem.
Vi =initial speed up = 12 sin 28 = 5.63 m/s
v = Vi - 9.81 t
at top v = 0
so
t = 5.63 /9.81 = 0.574 second
h=Hi + Vi t - (9.81/2) t^2 = 0 + 5.63*.574 - 4.9*.574^2
= 3.23 - 1.61 = 1.61 meters hi
If you wanted to know where it lands do constant horizontal velocity
u = 12 cos 28 forever
time =2 t = 2 * .574
range = 2 * .574 * 12 cos 28
= Vi cos
To determine the peak height of the long jumper, we need to analyze the motion in the vertical direction.
Step 1: Resolve the initial velocity into its vertical and horizontal components.
The vertical component of the initial velocity (Viy) is given by:
Viy = V * sin(θ)
= 12 m/s * sin(28°)
≈ 5.79 m/s
Step 2: Determine the time taken to reach the peak height.
The time taken (t) to reach the peak height can be calculated using the vertical component of the initial velocity and the acceleration due to gravity.
Using the kinematic equation:
Vf = Viy + gt
Where:
Vf = final velocity (0 at the peak)
Vi = initial velocity (Viy)
g = acceleration due to gravity (9.8 m/s^2)
0 = Viy + gt
0 = 5.79 m/s + (-9.8 m/s^2) * t
Solve for t:
9.8 m/s^2 * t = 5.79 m/s
t ≈ 0.59 s
Step 3: Calculate the peak height (h).
The peak height (h) can be calculated using the vertical component of the initial velocity and the time taken to reach the peak height.
Using the kinematic equation:
h = Viy * t + (1/2) * g * t^2
h = 5.79 m/s * 0.59 s + (1/2) * 9.8 m/s^2 * (0.59 s)^2
h ≈ 1.71 m
Therefore, the peak height of the long jumper is approximately 1.71 meters.
To determine the peak height of the long jumper, you can use the following steps:
Step 1: Resolve the initial velocity into its horizontal and vertical components.
- The horizontal component (Vx) of the initial velocity can be found using the formula Vx = V * cos(θ), where V is the initial velocity and θ is the angle above the horizontal.
- The vertical component (Vy) of the initial velocity can be found using the formula Vy = V * sin(θ), where V is the initial velocity and θ is the angle above the horizontal.
Step 2: Calculate the time taken to reach maximum height.
- The time taken to reach the maximum height can be found using the formula t = Vy / g, where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 3: Determine the maximum height.
- The maximum height can be calculated using the formula h = (Vy)^2 / (2 * g), where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's calculate the peak height of the long jumper using the given values:
- Initial velocity (V) = 12 m/s
- Angle above the horizontal (θ) = 28°
- Acceleration due to gravity (g) = 9.8 m/s^2
Step 1:
Vx = V * cos(θ) = 12 * cos(28°) ≈ 10.729 m/s
Vy = V * sin(θ) = 12 * sin(28°) ≈ 5.795 m/s
Step 2:
t = Vy / g = 5.795 / 9.8 ≈ 0.591 seconds
Step 3:
h = (Vy)^2 / (2 * g) = (5.795)^2 / (2 * 9.8) ≈ 1.705 meters
Therefore, the peak height of the long jumper is approximately 1.705 meters.