The muzzle velocity of a projectile fired from a gun has an upward component of 49 m/s and a horizontal component of 60 m/s.
A. What Maximum height does the projectile reach?
B. How far forward does it go (assume a level surface)
Show step by step and how you did it and explain.
A. To determine the maximum height reached by the projectile, we need to find the time it takes to reach the top of its path. From the given information, we know that the upward component of the muzzle velocity is 49 m/s.
Step 1: Split the projectile's motion into horizontal and vertical components.
- The vertical velocity at the highest point is zero since the projectile reaches its maximum height.
- The initial vertical velocity is 49 m/s.
Step 2: Use the kinematic equation to find the time taken to reach the top.
- We can use the equation: Vf = Vi + at, where Vf is the final velocity, Vi is the initial velocity, a is the acceleration and t is the time taken.
- Since the final vertical velocity is zero, we have 0 = 49 m/s - 9.8 m/s² * t. (The acceleration due to gravity is approximately 9.8 m/s².)
- Solving for t, we get t = 49 m/s / 9.8 m/s² = 5 seconds.
Step 3: Calculate the maximum height reached by the projectile.
- We can use the equation: d = Vi * t - 0.5 * a * t², where d is the displacement, Vi is the initial velocity, a is the acceleration and t is the time taken.
- Plugging in the values, we have d = 49 m/s * 5 s - 0.5 * 9.8 m/s² * (5 s)².
- Simplifying, we get d = 245 m - 0.5 * 9.8 m/s² * 25 s².
- Therefore, the maximum height reached by the projectile is d = 245 m - 122.5 m = 122.5 m.
So, the maximum height reached by the projectile is 122.5 meters.
B. To determine how far forward the projectile goes, we need to find the horizontal distance traveled. From the given information, we know the horizontal component of the muzzle velocity is 60 m/s.
Step 1: Use the formula for distance to calculate the horizontal distance traveled.
- The formula for distance is d = v * t, where d is the distance, v is the horizontal velocity, and t is the time taken.
- Plugging in the values, we have d = 60 m/s * 5 s.
- Therefore, the projectile travels d = 300 m forward.
So, the projectile goes 300 meters forward assuming a level surface.
To solve this problem, we can use the equations of motion for projectile motion. The horizontal and vertical components of motion are independent of each other, so we can analyze them separately.
A. To find the maximum height reached by the projectile, we need to determine the time it takes to reach the highest point of its trajectory. We can use the vertical component of the initial velocity and the acceleration due to gravity.
1. Given:
- Vertical component of initial velocity (upward) = 49 m/s
- Acceleration due to gravity (g) = 9.8 m/s^2 (assuming no air resistance)
2. The formula to calculate the time of flight for an object in projectile motion is:
Time = (Final Velocity - Initial Velocity) / Acceleration
In this case, the final vertical velocity is -49 m/s as the object reaches the maximum height and changes direction (going downwards). The initial vertical velocity is 49 m/s.
Time = (-49 m/s - 49 m/s) / (-9.8 m/s^2)
Time = -98 m/s / -9.8 m/s^2
Time = 10 s
3. Now that we know the time of flight, we can find the maximum height using the formula:
Height = Initial Velocity * Time + (0.5 * Acceleration * Time^2)
Height = 49 m/s * 10 s + (0.5 * -9.8 m/s^2 * (10 s)^2)
Height = 490 m - 490 m
Height = 0 m
Therefore, the projectile does not reach any maximum height.
B. To find how far forward the projectile goes assuming a level surface, we need to calculate the horizontal distance traveled using the horizontal component of its velocity and the time of flight.
1. Given:
- Horizontal component of initial velocity = 60 m/s
- Time of flight = 10 s (from part A)
2. The formula to calculate horizontal distance is:
Distance = Velocity * Time
Distance = 60 m/s * 10 s
Distance = 600 m
Therefore, the projectile travels 600 meters horizontally on a level surface.
To solve this problem, we need to understand the basic principles of projectile motion. A projectile is an object moving through the air that is subject only to the forces of gravity and air resistance (which we will neglect in this case). The motion of a projectile can be broken down into two independent components: horizontal and vertical.
Given:
- Upward component of velocity (vertical) = 49 m/s
- Horizontal component of velocity = 60 m/s
Now, let's solve the problem step by step:
A. Maximum height of the projectile:
1. The vertical component of the velocity affects the height of the projectile. At the maximum height, the vertical component of velocity becomes zero.
2. The motion of the projectile can be divided into two parts: upward and downward.
Step 1: Find the time taken for the projectile to reach its maximum height.
- We can use the kinematic equation: v = u + at, where
- v = final velocity (0 m/s at maximum height)
- u = initial velocity (49 m/s upwards)
- a = acceleration (acceleration due to gravity, -9.8 m/s²)
- t = time
0 = 49 - 9.8t
9.8t = 49
t = 49 / 9.8
t ≈ 5 seconds
Step 2: Find the maximum height using the time obtained.
- We can use the kinematic equation: s = ut + 0.5at², where
- s = displacement (maximum height)
- u = initial velocity (49 m/s upwards)
- t = time (5 seconds)
- a = acceleration (acceleration due to gravity, -9.8 m/s²)
s = (49)(5) + 0.5(-9.8)(5)²
s = 245 + 0.5(-9.8)(25)
s = 245 - 122.5
s ≈ 122.5 meters
Therefore, the maximum height the projectile reaches is approximately 122.5 meters.
B. Distance traveled by the projectile:
1. The horizontal component of velocity affects the distance traveled by the projectile. It remains constant throughout the motion.
Step 1: Find the time of flight of the projectile.
- The time of flight is the total time taken for the projectile to reach the ground.
- The vertical distance covered during this time is the same as the distance covered horizontally.
We already found that the time of flight is approximately 5 seconds.
Step 2: Calculate the horizontal distance.
- We can use the formula: distance = velocity × time, where
- distance = horizontal distance traveled by the projectile
- velocity = horizontal component of velocity (60 m/s)
- time = time of flight (5 seconds)
distance = 60 × 5
distance = 300 meters
Therefore, the projectile travels approximately 300 meters forward on a level surface.
By following the steps above and using the concepts of projectile motion, we can accurately determine the maximum height and horizontal distance traveled by the projectile.
In y
max height:
y = v^2/g
time:
t = 2v/g
In x
x = vt (you found from y)