Projectile Motion

 An object is thrown straight up from the top of a building h feet tall with an initial velocity of v feet per second. The height of the object as a function of time can be modeled by the function h(t) = –16t2 + vt + h, where h(t) is the height of the object (in feet) t seconds after it is thrown. If we are given the initial velocity (or speed) of the object and the height of the building, we can use this model to determine how long it takes for the object to reach various heights. This model assumes that the object misses the top of the building on the way back down to the ground and that wind resistance is minimal. Here are the steps required for Solving Projectile Motion Problems:

 Step 1: Set the given equation equal to the appropriate height. Step 2: Solve the equation found in step 1 by setting the equation equal to zero and factoring the equation. Step 3: Based on the problem, determine which answer or answers are correct. Do not forget to include the units in your final answer.

Example 1 – A ball is thrown straight up from the top of a 128 foot tall building with an initial speed of 32 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t2 + 32t + 128. How long will it take for the ball to hit the ground?

 Step 1: Set the given equation equal to the appropriate height. In this case, we set the equation equal to zero because the height of the ground is zero. Step 2: Solve the equation found in step 1 by setting the equation equal to zero and factoring the equation. Step 3: Based on the problem, determine which answer or answers are correct. Do not forget to include the units in your final answer. In this case, there is only one positive answer which makes sense because the ball will only strike the ground once. Example 2 – A ball is thrown straight up from the top of a 288 foot tall building with an initial speed of 48 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t2 + 48t + 288. When will the ball reach a height of 320 feet?

 Step 1: Set the given equation equal to the appropriate height. In this case, we set the equation equal to 320 because we want to determine when the height will be 320 feet. Step 2: Solve the equation found in step 1 by setting the equation equal to zero and factoring the equation. Step 3: Based on the problem, determine which answer or answers are correct. Do not forget to include the units in your final answer. In this case, there are two positive answers which makes sense because the ball will reach 320 feet once on the way up and again on the way down. Example 3 – A rocket is launched straight up from the top of a 24 foot tall building with an initial speed of 92 feet per second. The height of the rocket as a function of time can be modeled by the function h(t) = –16t2 + 92t + 24. How long will it take for the rocket to hit the ground?

 Step 1: Set the given equation equal to the appropriate height. In this case, we set the equation equal to zero because the height of the ground is zero. Step 2: Solve the equation found in step 1 by setting the equation equal to zero and factoring the equation. Step 3: Based on the problem, determine which answer or answers are correct. Do not forget to include the units in your final answer. In this case, there is only one positive answer which makes sense because the ball will only strike the ground once. Example 4 – A ball is hit straight up in the air from a height of 4 feet with an initial speed of 64 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t2 + 64t + 4. When will the ball reach a height of 52 feet?

 Step 1: Set the given equation equal to the appropriate height. In this case, we set the equation equal to 52 because we want to determine when the height will be 52 feet. Step 2: Solve the equation found in step 1 by setting the equation equal to zero and factoring the equation. Step 3: Based on the problem, determine which answer or answers are correct. Do not forget to include the units in your final answer. In this case, there are two positive answers which makes sense because the ball will reach 52 feet once on the way up and again on the way down. 