# A projectile is shot from the ground at an angle of #pi/4 # and a speed of #3 m/s#. Factoring in both horizontal and vertical movement, what will the projectile's distance from the starting point be when it reaches its maximum height?

The distance is

We apply the equation of motion

to calculate the time to reach the greatest height

To find the distance, we apply the equation of motion

By signing up, you agree to our Terms of Service and Privacy Policy

To find the distance from the starting point when the projectile reaches its maximum height, we first need to determine the time it takes to reach the maximum height. This occurs when the vertical component of the projectile's velocity becomes zero. The time to reach maximum height can be found using the formula:

[ t = \frac{v_{y0}}{g} ]

Where: ( v_{y0} ) = initial vertical velocity ( g ) = acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 ))

Given that the projectile is launched at an angle of ( \frac{\pi}{4} ) radians with a speed of ( 3 , \text{m/s} ), the initial vertical velocity (( v_{y0} )) can be calculated using trigonometric functions:

[ v_{y0} = v_0 \sin(\theta) ]

Substitute the values:

[ v_{y0} = 3 , \text{m/s} \times \sin\left(\frac{\pi}{4}\right) = 3 , \text{m/s} \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} , \text{m/s} ]

[ t = \frac{\frac{3\sqrt{2}}{2}}{9.8} \approx 0.306 , \text{s} ]

Now, to find the horizontal distance traveled at the maximum height, we use the time calculated:

[ d_{x\text{ max}} = v_{x0} \times t ]

Where: ( v_{x0} ) = initial horizontal velocity

Given that ( v_{x0} = v_0 \cos(\theta) ):

[ v_{x0} = 3 , \text{m/s} \times \cos\left(\frac{\pi}{4}\right) = 3 , \text{m/s} \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} , \text{m/s} ]

[ d_{x\text{ max}} = \frac{3\sqrt{2}}{2} , \text{m/s} \times 0.306 , \text{s} \approx 0.459 , \text{m} ]

So, the projectile's distance from the starting point when it reaches its maximum height is approximately ( 0.459 , \text{m} ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the projection of #(-i + j + k)# onto # ( i - j + k)#?
- How can acceleration change the motion of an object?
- How do you normalize # (5i- 3j + 12k) #?
- In projectile motion, why does an object that is vertically dropped down and an object that is horizontally launched reach the ground at the same time?
- What is the cross product of #<4 , 5 ,-9 ># and #<4, 3 ,0 >#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7