Assuming no air resistance or forces other than the Earth's gravity, the height above the ground at time tt (in seconds) of a falling object is given by s(t)=−4.9t2+v0t+s0s(t)=-4.9t2+v0t+s0, where ss is in meters, v0v0 is the object's initial velocity (in msms), and s0s0 is the initial height of the object (in mm).
Now suppose the object is propelled upwards from the ground s0=0s0=0 with an intitial velocity of 56 msms.    
   **2)** When does the object hit the ground? 

To determine when the object hits the ground, we need to find the time \( t \) when the height \( s(t) \) is equal to zero, which represents the object reaching the ground.

Given the equation for height:

\[
s(t) = -4.9t^2 + v_0 t + s_0
\]

We know that:

- The initial velocity \( v_0 = 56 \, \text{m/s} \),
- The initial height \( s_0 = 0 \, \text{m} \) (since the object is launched from the ground).

Thus, the equation simplifies to:

\[
s(t) = -4.9t^2 + 56t
\]

Now, to find when the object hits the ground, we set \( s(t) = 0 \):

\[
0 = -4.9t^2 + 56t
\]

This is a quadratic equation. Factoring out \( t \):

\[
t(-4.9t + 56) = 0
\]

This gives two solutions:

1. \( t = 0 \) (the object is on the ground at the initial time),
2. \( -4.9t + 56 = 0 \).

Solving for \( t \):

\[
-4.9t = -56
\]
\[
t = \frac{56}{4.9} \approx 11.43 \, \text{seconds}.
\]

Therefore, the object hits the ground at approximately \( t = 11.43 \, \text{seconds} \).

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Would you like more details or have any questions?

Here are five questions that build upon this topic:
1. How would the time to hit the ground change if the initial velocity were doubled?
2. What is the maximum height the object reaches before falling?
3. How would the time to hit the ground change if the object were dropped instead of propelled upwards?
4. How does the height equation change if we include air resistance?
5. Can you derive the equation for the object's velocity as a function of time?

**Tip:** Always pay attention to the signs in kinematic equations. Positive velocity means upward motion, and negative velocity means downward motion.

When does the object hit the ground if it is propelled upwards from the ground with an initial velocity of 56 m/s?

Learn how to solve a kinematics problem where an object is propelled upwards from the ground with an initial velocity of 56 m/s. Using the quadratic equation and kinematic principles, we determine the time when the object hits the ground. Suitable for high school students, grades 9-12.

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