Physics 16 How Far Does a Projectile Go?

When a projectile such as a ball is tossed or thrown, we saw the ball has two big vectors. One is gravity pulling the ball downward. The other is the force pushing the ball forward.

Finding how far a projectile goes is influenced by both of these vectors. But how?

Try this experiment [a friend observing helps]: Stand still and drop a ball catching it on the first bounce. Do not throw it. Ask your friend the path the ball followed down and up.

movement of a dropped ball graphic

A dropped ball goes straight down due to gravity then bounces straight up due to elasticity of the ball.

Next walk across the floor. As you walk, drop the ball and catch it on the first bounce. Ask your friend about the path this ball followed.

When you were standing still, the ball had to go straight down and up for you to catch it. When you were walking, if the ball dropped straight down and came straight up, you could not catch it because you were moving. The ball had to follow a curved or projectile pathway.

dropping a ball while walking graphic

When you walk forward, you have momentum. Even when you just drop a ball, it has some of the momentum from you so ti follows a curved path down and up so you can catch it instead of passing it by.

This shows the two movements of a projectile, although influenced by each other, work independently. This is important for finding out how far a projectile will go.

materials needed for Project

Question: How far does a projectile go?

Materials:

Rubber ball

Meter stick

Stopwatch

Procedure:

Repeat dropping and catching the ball as you walk to be sure you are dropping the ball not dribbling it. The only downward force is supposed to be gravity.

Measure the distance from where you will drop the ball to the floor

Mark a starting line on the floor. You will have to be walking as you cross this line so give yourself some room.

setting up the walking path

The meter stick will probably be long enough to show how far you walk dropping and catching the ball. Set is up from your starting line along the path you will follow.

Put the meter stick down with one end touching the starting line

Holding the ball, start walking toward the starting line

As you cross the starting line, drop the ball and start the stopwatch

Stop, stop the stopwatch when you catch the ball on the first bounce

Find the distance you went by looking at the meter stick

[You may want to have your friend work the stopwatch.]

Doing this can be difficult so you may want to repeat it a few times

Observations:

Describe the path the ball follows when you drop it:

Describe the path the ball follows when you drop it while walking:

Distance the ball is dropped:

For each trial with the ball write down the time and distance:

Analysis:

The falling projectile ball has two different movements so there are two different vectors.

vectors showing how the curved path happens

When you drop a ball while walking, ti follows a curved path. In vectors this is shown with two vectors, one going forward for the momentum and one going down for the gravity. Added up the two show where the ball will land.

Vector 1 – Distance

One of the vectors points forward because you are moving forward. How far forward will the ball go?

We’ve seen this already. The distance is the time multiplied by the speed. You want to know about the part when the ball is going to the floor. As we saw in the last Project, this is half of the total. Divide the distance and the time in half.

Now you can rearrange the formula to v = d/t. Put in your values for d and t to find the velocity for the first vector.

Vector 2 – Falling

We’ve seen this already. Gravity is acceleration so the distance down equals gravity’s acceleration multiplied by the square of the time.

Remember the acceleration is 9.8 m/s2 and the time is what you used before. Put your values in and calculate the distance the ball dropped.

Now compare the distance you dropped the ball to what you calculated.

Looking at What Makes Your Values Change

We used several measurements in our calculations. One was the value for gravity. We used an accepted and proven value of 9.8 m/s2 so this will not change.

vectors for dropping a ball while walking

There are three measurements you need for a projectile. One is time. One is the distance to the ground. The third is the distance from dropping to catching the ball.

Now we have several measurements we made. One was the distance you walked between dropping and catching the ball. We assumed we dropped and caught the ball at the same height.

How accurately did you measure the distance? It must be in meters. If you measured the distance and the time in different tests, can you be sure the distance was the same both times?

Another distance was how far you dropped the ball. When should we have measured this distance? Why? Is there a way to make sure you drop the ball from the same height each time?

We also measured the time it took to bounce the ball. My ball bounced up very fast making it hard to time this accurately. What about your time?

In the last Project we threw a ball upwards. The force we threw it with was countered by gravity until the two were equal and gravity took over causing the ball to fall down. Half the time was spent going up and half coming down.

When we drop a ball, gravity pulls it down. When the ball bounces, the force of the ball hitting the ground makes it come back up. Different kinds of balls will bounce differently.

What kind of ball did you use? I used one of those rubber balls that bounces back as high as the height I dropped it from for the first bounce. This kind of ball would come very close to taking the same amount of time coming up as it did going down.

If you used a different kind of ball, this may not be true. That would mean your time value would not be right. That would make your calculations off too.

Taking Another Look At How This Works

A projectile differs from a falling object because it moves from a starting to an ending place. The projectile or ball we looked at went out horizontally and fell as it traveled across this distance.

The arc of the falling projectile has two vector parts. One is falling because of gravity. The other is going forward due to some force pushing it forward.

With the balls we dropped, what was the forward push? When we walk forward, we have something called momentum. We will look at this more closely in a future Project. This momentum provided the force forward to our balls.

Calculating the time it takes for the ball to fall to the ground uses the formula d = at2 where a is 9.8 m/s2 for gravity and d is the distance we measure.

Calculating how far forward the ball will go uses the formula d = vt. If we know how far the ball will fall, we can calculate time as above or we can measure it.

We must know the distance to calculate velocity or velocity to calculate distance. We must know two values for the formula to calculate the third.

Giving It another Try

This time let’s try to keep our measurements much more accurate. The first one is how far we will drop the ball.

The farther the ball drops, the longer it will take making timing easier. However, we need to have some mark so we drop it and catch it at the same height. The waist is convenient but may make timing difficult.

Measure the height to your waist.

Next is the distance you walk while bouncing the ball.You need to be walking along before you drop the ball. Have a route and a marked starting line. It may help to have a line or something marked to keep walking in a straight line. Have a way to mark the ending spot.

The hardest measurement is the time. The ball will bounce back up quickly. Having a friend time the bounce may be easier than trying to walk a straight line, drop the ball while starting the stopwatch and catching the ball as you stop the stopwatch.

You can do several trials but keep each set of measurements separate as each trial may differ in  measurements from the others. They should be similar so you can pick a set that seems more accurate to use for your calculations.

What Can You Calculate?

You can use your distance and time measurements to calculate your walking velocity using v = d/t.

You can use d = at2 where a = 9.8 m/s2 to calculate the distance the ball falls to the ground or the time it takes to reach the ground. Because the time is squared, you will want to use a calculator to find the square root or the time that was squared.

What We Will Do Next

Next week the Project will be a projectile challenge. We will calculate the velocity of a ball traveling off a table and use it to calculate where to place a cup to catch the ball when it reaches the ground. You will need two long ramps, a ball, a meter stick and a stopwatch. Oh, you will need a cup to catch the ball in.