A hockey puck glides across the ice at 27.7 m/s, when a player whacks it with her hockey stick, giving it an acceleration at 64.3∘ to its original direction. The acceleration lasts 50.3 ms, and the puck's displacement during this time is 1.60 m.
To find the initial velocity of the hockey puck, you can use the equation:
v = u + at
Where:
v = final velocity (27.7 m/s)
u = initial velocity (unknown)
a = acceleration (unknown)
t = time (50.3 ms or 0.0503 s)
Since the acceleration is at an angle of 64.3° to the original direction, we need to break it down into its vertical and horizontal components. The horizontal component of acceleration can be found using the equation:
a_horizontal = a * cosθ
Where:
a_horizontal = horizontal component of acceleration
a = total acceleration (unknown)
θ = angle of acceleration (64.3°)
Substituting the given values into the equation:
a_horizontal = a * cos(64.3°)
Next, we can find the vertical component of acceleration using the equation:
a_vertical = a * sinθ
Where:
a_vertical = vertical component of acceleration
Substituting the given values into the equation:
a_vertical = a * sin(64.3°)
During the time interval of 0.0503 seconds, the displacement of the puck can be found using the equation:
s = ut + (1/2)at^2
Where:
s = displacement (1.60 m)
u = initial velocity (unknown)
t = time (0.0503 s)
a = total acceleration (unknown)
Since the acceleration has both horizontal and vertical components, we need to consider them separately and calculate the total acceleration later.
For the vertical motion:
s_vertical = u_vertical * t + (1/2) * a_vertical * t^2
Where:
s_vertical = vertical displacement
u_vertical = vertical component of initial velocity
a_vertical = vertical component of acceleration
For the horizontal motion:
s_horizontal = u_horizontal * t + (1/2) * a_horizontal * t^2
Where:
s_horizontal = horizontal displacement
u_horizontal = horizontal component of initial velocity
a_horizontal = horizontal component of acceleration
Relating the horizontal and vertical displacements to the total displacement, we can use the Pythagorean theorem:
s^2 = s_horizontal^2 + s_vertical^2
You can now solve these equations simultaneously to find the initial velocity and total acceleration of the hockey puck.