An object on the ground is launched at 34 m/s at an angle of 64 degrees above the horizontal. If the object lands on top of a 4.5 m tall hill, then what is the horizontal displacement of the object?
To find the horizontal displacement of the object, we need to analyze the projectile motion. Here's how you can determine the horizontal displacement:
1. Break down the initial velocity into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component is affected by gravity.
Horizontal component: v₀x = v₀ * cos(θ)
Vertical component: v₀y = v₀ * sin(θ)
Given that the initial velocity (v₀) is 34 m/s and the launch angle (θ) is 64 degrees, you can calculate the horizontal and vertical components as follows:
v₀x = 34 m/s * cos(64°)
v₀y = 34 m/s * sin(64°)
2. Determine the time of flight (t) for the object. The time it takes for the object to reach its highest point and then descend is equal to 2 times the duration it takes for the object to reach its maximum height.
For an object launched at an angle above the horizontal, the time of flight can be calculated using the following formula:
t = (2 * v₀y) / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the values:
t = (2 * v₀ * sin(θ)) / g
3. Calculate the horizontal displacement (d) of the object. The horizontal displacement can be determined by multiplying the horizontal component of velocity (v₀x) by the time of flight (t).
d = v₀x * t
Substituting the values:
d = (v₀ * cos(θ)) * t
4. Calculate the time taken for the object to reach the top of the hill (t_hill). The object's vertical displacement at this point is the height of the hill (4.5 m). Use the following equation to calculate the time taken:
4.5 m = v₀y * t_hill - (1/2) * g * t_hill²
Solve this quadratic equation for t_hill. You may need to use the quadratic formula.
Once you have the time taken to reach the top of the hill, subtract this time from the total time of flight to get the time taken for the object to reach the ground (t_ground).
5. Finally, calculate the horizontal displacement (d_final) of the object after it lands on top of the hill. This can be determined by multiplying the horizontal component of velocity (v₀x) by the time taken to reach the ground (t_ground).
d_final = v₀x * t_ground
By following these steps and performing the calculations, you will be able to determine the horizontal displacement of the object.