An athlete executing a long jump, leaves the ground at a 30 degree angle with the ground and travels 8.90 meters horizontally in 1.1 seconds. What was the takeoff speed of the athlete along the diagonal?
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To find the takeoff speed of the athlete along the diagonal, we need to break down the given information and use the principles of projectile motion.
First, let's break down the horizontal and vertical components of the athlete's motion:
- The horizontal distance traveled, 8.90 meters, will help us find the horizontal component of the velocity.
- The angle of takeoff, 30 degrees, will help us find the vertical and diagonal components of the velocity.
Let's start by finding the horizontal component of the velocity, which remains constant throughout the motion. We can use the formula:
Horizontal distance = Horizontal velocity × Time
Plugging in the values, we get:
8.90 meters = Horizontal velocity × 1.1 seconds
To find the horizontal velocity, we rearrange the equation:
Horizontal velocity = 8.90 meters / 1.1 seconds
Calculating this, we get:
Horizontal velocity = 8.09 meters per second
Now that we have the horizontal velocity, we can find the diagonal and vertical components of the velocity using trigonometry.
The diagonal component of the velocity is given by:
Diagonal velocity = Horizontal velocity / cosine(angle of takeoff)
Plugging in the values, we have:
Diagonal velocity = 8.09 meters per second / cosine(30 degrees)
Calculating this, we get:
Diagonal velocity = 9.34 meters per second
Therefore, the takeoff speed of the athlete along the diagonal is approximately 9.34 meters per second.