A projectile of mass 0.636 kg is shot from a cannon, at height 6.6 m, as shown in the figure, with an initial velocity vi having a horizontal component of 6.8 m/s. The projectile rises to a maximum height of ∆y above the end of the cannon’s barrel and strikes the ground a horizontal distance ∆x
past the end of the cannon’s barrel.Find the magnitude of the velocity vector
when the projectile hits the ground. The
acceleration of gravity is 9.8 ms2. Answer in units of m/s.
7.7 m/s
To find the magnitude of the velocity vector when the projectile hits the ground, we can use the principles of projectile motion.
Step 1: Find the time of flight
The time of flight (T) is the total time taken by the projectile to reach the ground. We can calculate it using the following formula:
T = (2 * Δy / g)^0.5
where Δy is the maximum height reached by the projectile and g is the acceleration due to gravity.
Given Δy = 6.6 m and g = 9.8 m/s^2, we can substitute these values into the formula:
T = (2 * 6.6 / 9.8)^0.5
Step 2: Find the horizontal distance traveled
The horizontal distance traveled by the projectile can be calculated by:
Δx = vi * T
where vi is the initial horizontal component of the velocity.
Given vi = 6.8 m/s (horizontal component) and T from Step 1, we can substitute these values into the formula:
Δx = 6.8 * T
Step 3: Find the final vertical velocity
The final vertical velocity (vf) can be calculated using the formula:
vf = g * T
Given g = 9.8 m/s^2, and T from Step 1, we can substitute these values into the formula:
vf = 9.8 * T
Step 4: Find the resultant velocity
The resultant velocity (v) can be calculated using the formula:
v = (vf^2 + vi^2)^0.5
Given vf from Step 3 and vi = 6.8 m/s, we can substitute these values into the formula:
v = (vf^2 + 6.8^2)^0.5
Finally, we substitute the values for vf from Step 3 into the above equation and calculate the magnitude of the velocity vector when the projectile hits the ground.
To find the magnitude of the velocity vector when the projectile hits the ground, we can apply the principles of projectile motion. Here are the steps to solve the problem:
1. Determine the time it takes for the projectile to reach the maximum height (∆y):
- The initial vertical velocity component (viy) can be calculated using the given initial velocity (vi) and the angle of projection (θ).
- The time taken to reach the maximum height can be calculated using the formula: t = viy / g, where g is the acceleration due to gravity.
2. Calculate the maximum height (∆y) reached by the projectile:
- Use the formula: ∆y = (viy)^2 / (2 * g).
- Substitute the values of viy and g to get the value of ∆y.
3. Determine the time of flight (total time in the air):
- The time of flight is twice the time taken to reach the maximum height.
- Multiply the time taken to reach the maximum height by 2 to get the total time in the air.
4. Calculate the horizontal distance (∆x) traveled by the projectile:
- Use the formula: ∆x = vix * t, where vix is the horizontal component of the initial velocity and t is the time of flight.
5. Determine the final vertical velocity component (vf):
- Use the formula: vf = sqrt((viy)^2 + 2 * g * ∆y).
6. Calculate the final velocity (v) using the Pythagorean theorem:
- The final velocity is the magnitude of the velocity vector and can be calculated using the formula: v = sqrt((vix)^2 + (vf)^2).
By following these steps and substituting the given values into the equations, you can find the magnitude of the velocity vector when the projectile hits the ground.