Part 1:
A long jumper leaves theground at an angle of 21.7 degrees to the horizontal and at a speed of 10.6 m/s.
The acceleration of gravity is 9.8 m/s^2. How far does he jump? Answer in units of m.
Part 2:
What is the maximum height he reaches? Answer in units of m.
To solve this problem, we need to break it down into two parts: analyzing the horizontal motion and analyzing the vertical motion.
Part 1: Horizontal Motion
The horizontal motion of the long jumper is independent of his vertical motion and is unaffected by gravity. Therefore, we can calculate the horizontal distance traveled using the formula:
Distance = Speed × Time
The long jumper's speed is given as 10.6 m/s, and we need to find the time. From the given information, we know the initial vertical velocity is 0 m/s when the long jumper leaves the ground. So, we can assume that the time taken for the jumper to reach the highest point is equal to the time taken for him to return to the ground.
To find the time of flight, we can use the equation:
Time = (2 × Initial Vertical Velocity) / Acceleration due to gravity
Plugging in the values, we get:
Time = (2 × 10.6 sin(21.7°)) / 9.8
Now, we can calculate the horizontal distance traveled by multiplying the time of flight by the horizontal velocity:
Distance = Speed × Time
Substituting the values, we get:
Distance = 10.6 cos(21.7°) × [(2 × 10.6 sin(21.7°)) / 9.8]
Simplifying this expression will give you the distance the long jumper jumps in meters.
Part 2: Vertical Motion
To find the maximum height the long jumper reaches, we need to calculate the vertical component of the initial velocity and use the kinematic equation:
Final Vertical Velocity^2 = Initial Vertical Velocity^2 + 2 × Acceleration due to gravity × Vertical Displacement
At the highest point of the jump, the final vertical velocity is 0 m/s. We already know the initial vertical velocity and the acceleration due to gravity. Now we need to find the vertical displacement.
Using the equation:
Vertical Displacement = (Initial Vertical Velocity^2) / (2 × Acceleration due to gravity)
Substituting the values, we get:
Vertical Displacement = (10.6 sin(21.7°))^2 / (2 × 9.8)
This will give you the maximum height reached by the long jumper in meters.