Enter the final height, time, and initial vertical speed (then choose up/down) into the calculator to determine the initial height from which an object was dropped or thrown vertically. This calculator assumes the only force acting on the object is gravity (air resistance is ignored).
Initial Height Formula
The following formula is used to calculate the initial height (using the convention that up is positive and g is the positive magnitude of gravitational acceleration):
H_i = H_f - V_0 * t + 0.5 * g * t^2
Variables:
- H_i is the initial height (meters)
- H_f is the final height (meters)
- V_0 is the initial vertical velocity (meters per second; positive upward, negative downward)
- t is the time (seconds)
- g is the acceleration due to gravity magnitude (standard Earth value 9.80665 m/s²; acceleration is downward, so a = −g in this sign convention)
To calculate the initial height, subtract the product of the initial velocity and time from the final height, then add half of g times the square of the time (with the “up is positive” sign convention).
What is Initial Height?
Initial height is the height from which an object starts its motion when dropped or thrown vertically upwards or downwards. It is an important parameter in kinematics for determining the motion of objects under the influence of gravity.
How to Calculate Initial Height?
The following steps outline how to calculate the Initial Height.
- First, determine the final height (H_f) in meters.
- Next, determine the initial vertical velocity (V_0) in meters per second and its direction (up positive, down negative).
- Next, determine the time (t) in seconds.
- Use the formula H_i = H_f − V_0 * t + 0.5 * g * t^2 to calculate the initial height (H_i).
- After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem:
Use the following variables as an example problem to test your knowledge.
final height (H_f) = 20 meters
initial vertical velocity (V_0) = 5 meters per second upward
time (t) = 3 seconds
Using g = 9.80665 m/s²: H_i = 20 − (5)(3) + 0.5(9.80665)(3²) ≈ 49.1299 meters.
